[PAS] Probability Abstracts 97
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pas at lists.imstat.org
Wed May 2 11:30:58 CDT 2007
Probability Abstracts 97
This document contains abstracts 5305-5549 from
March-1-2007 to Apr-30-2007.
They have been mailed on May 2nd, 2007.
This letter can be also found on line at
http://lists.imstat.org/PAS/Letters/letter_97.shtml
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5305. QUADRATIC BSDES WITH RANDOM TERMINAL TIME AND ELLIPTIC PDES IN
INFINITE DIMENSION
Philippe Briand and Fulvia Confortola
In this paper we study one dimensional backward stochastic differential
equations (BSDEs) with random terminal time not necessarily bounded
or finite
when the generator F(t,Y,Z) has a quadratic growth in Z. We provide
existence
and uniqueness of a bounded solution of such BSDEs and, in the case
of infinite
horizon, regular dependence on parameters. The obtained results are then
applied to prove existence and uniqueness of a mild solution to elliptic
partial differential equations in Hilbert spaces.
http://arxiv.org/abs/0704.1223
---------------------------------------------------------------
5306. INTERSECTION LOCAL TIME FOR TWO INDEPENDENT FRACTIONAL BROWNIAN
MOTIONS
David Nualart and Salvador Ortiz-Latorre
We prove the existence of the intersection local time for two
independent, d
-dimensional fractional Brownian motions with the same Hurst
parameter H.
Assume d greater or equal to 2, then the intersection local time
exists if and
only if Hd<2.
http://arxiv.org/abs/0704.1259
---------------------------------------------------------------
5307. INTEGRAL FORMULAS FOR THE ASYMMETRIC SIMPLE EXCLUSION PROCESS
Craig A. Tracy and Harold Widom
In this paper we obtain general integral formulas for probabilities
in the
asymmetric simple exclusion process (ASEP) on the integer lattice
with nearest
neighbor hopping rates p to the right and q=1-p to the left. For the
most part
we consider an N-particle system but for certain of these formulas we
can take
the limit as N goes to infinity. First we obtain, for the N-particle
system, a
formula for the probability of a configuration at time t, given the
initial
configuration. For this we use Bethe Ansatz ideas to solve the master
equation,
extending a result of Schuetz for the case N=2. The main results of
the paper,
derived from this, are integral formulas for the probability, for
given initial
configuration, that the m'th left-most particle is at x at time t. In
one of
these formulas we can take the limit as N goes to infinity, and it
gives the
probability for an infinite system where the initial configuration is
bounded
on one side. For the special case of the totally asymmetric simple
exclusion
process (TASEP) our formulas reduce to the known ones.
http://arxiv.org/abs/0704.2633
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5308. DETERMINING FACTORS BEHIND THE PAGERANK LOG-LOG PLOT
Yana Volkovich and Nelly Litvak and Debora Donato
We study the relation between PageRank and other parameters of
information
networks such as in-degree, out-degree, and the fraction of dangling
nodes. We
model this relation through a stochastic equation inspired by the
original
definition of PageRank. Further, we use the theory of regular
variation to
prove that PageRank and in-degree follow power laws with the same
exponent. The
difference between these two power laws is in a multiple coefficient,
which
depends mainly on the fraction of dangling nodes, average in-degree,
the power
law exponent, and damping factor. The out-degree distribution has a
minor
effect, which we explicitly quantify. Our theoretical predictions
show a good
agreement with experimental data on three different samples of the Web.
http://arxiv.org/abs/0704.2694
---------------------------------------------------------------
5309. THE DYNAMICAL DISCRETE WEB
L. R. G. Fontes and C. M. Newman and K. Ravishankar and E. Schertzer
The dynamical discrete web (DDW), introduced in recent work of Howitt
and
Warren, is a system of coalescing simple symmetric one-dimensional
random walks
which evolve in an extra continuous dynamical parameter s. The
evolution is by
independent updating of the underlying Bernoulli variables indexed by
discrete
space-time that define the discrete web at any fixed s. In this
paper, we study
the existence of exceptional (random) values of s where the paths of
the web do
not behave like usual random walks and the Hausdorff dimension of the
set of
such exceptional s. Our results are motivated by those about
exceptional times
for dynamical percolation in high dimension by H\"aggstrom, Peres and
Steif,
and in dimension two by Schramm and Steif. The exceptional behavior
of the
walks in DDW is rather different from the situation for dynamical
random walks
of Benjamini, H\"aggstrom, Peres and Steif. In particular, we prove
that there
are exceptional values of s for which the walk from the origin S^s(n)
has
limsup S^s(n)/\sqrt n \leq K with a nontrivial dependence of the
Hausdorff
dimension on K. We also discuss how these and other results extend to
the
dynamical Brownian web, a natural scaling limit of DDW. The scaling
limit is
the focus of a paper in preparation; it was studied by Howitt and
Warren and is
related to the Brownian net of Sun and Swart.
http://arxiv.org/abs/0704.2706
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5310. MULTIDIMENSIONAL SDE WITH ANTICIPATING INITIAL PROCESS AND
REFLECTION
Zongxia Liang
In this paper, the strong solutions $ (X, L)$ of multidimensional
stochastic
differential equations with reflecting boundary and possible
anticipating
initial random variables is established. The key is to obtain some
substitution
formula for Stratonovich integrals via a uniform convergence of the
corresponding Riemann sums and to prove continuity of functionals of
$ (X, L)$.
http://arxiv.org/abs/0704.2715
---------------------------------------------------------------
5311. THE ORDER OF THE DECAY OF THE HOLE PROBABILITY FOR GAUSSIAN
RANDOM SU(M+1) POLYNOMIALS
Scott Zrebiec
We show that for Gaussian random SU(m+1) polynomials of a large
degree N the
probability that there are no zeros in the disk of radius r is less than
$e^{-c_{1,r} N^{m+1}}$, and is also greater than $e^{-c_{2,r} N^{m+1}}$.
Enroute to this result, we also derive a more general result:
probability
estimates for the event where the volume of the zero set of a random
polynomial
of high degree deviates significantly from its mean.
http://arxiv.org/abs/0704.2733
---------------------------------------------------------------
5312. TAMED 3D NAVIER-STOKES EQUATION: EXISTENCE, UNIQUENESS AND
REGULARITY
Michael R\"ockner and Xicheng Zhang
In this paper, we prove the existence and uniqueness of a smooth
solution to
a tamed 3D Navier-Stokes equation in the whole space. In particular,
if there
exists a bounded smooth solution to the classical 3D Navier-Stokes
equation,
then this solution satisfies our tamed equation. Moreover, using this
renormalized equation we can give a new construction for a suitable weak
solution of the classical 3D Navier-Stokes equation introduced in
[Scheffer:
Hausdorff measure and the Navier-Stokes equations. Comm. Math. Phys.,
1977] and
[Caffarelli, Kohn, Nirenberg: Partial regularity of suitable weak
solutions of
the Navier-Stokes equations. Comm. Pure Appl. Math., 1982].
http://arXiv.org/abs/math/0703254
---------------------------------------------------------------
5313. ON STOCHASTIC EVOLUTION EQUATIONS WITH NON-LIPSCHITZ COEFFICIENTS
Xicheng Zhang
In this paper, we study the existence and uniqueness of solutions for
several
classes of stochastic evolution equations with non-Lipschitz
coefficients, that
is, backward stochastic evolution equations, stochastic Volterra type
evolution
equations and stochastic functional evolution equations. In
particular, the
results can be used to treat a large class of quasi-linear stochastic
equations, which includes the reaction diffusion and porous medium
equations.
http://arXiv.org/abs/math/0703260
---------------------------------------------------------------
5314. LARGE DEVIATIONS FOR RANDOM WALKS UNDER SUBEXPONENTIALITY: THE
BIG-JUMP DOMAIN
D. Denisov and A. B. Dieker and V. Shneer
For a given one-dimensional random walk {S_n} with a subexponential
step-size
distribution, we present a unifying theory to study the sequences
{x_n} for
which P{S_n>x} \sim n P{S_1>x} as n\to\infty uniformly for x\ge x_n.
We also
investigate the stronger `local' analogue, P{S_n\in(x,x+T]}\sim n
\pr{S_1\in(x,x+T]}. Our theory is self-contained and fits well within
classical
results on domains of (partial) attraction and local limit theory.
When specialized to the most important subclasses of subexponential
distributions that have been studied in the literature, we reproduce
known
results. Importantly, we supplement these well-known theorems with
new results.
http://arXiv.org/abs/math/0703265
---------------------------------------------------------------
5315. RATE OF GROWTH OF A TRANSIENT COOKIE RANDOM WALK
Anne-Laure Basdevant (PMA) and Arvind Singh (PMA)
We consider a one-dimensional transient cookie random walk. It is
known from
a previous paper that a cookie random walk $(X_n)$ has positive or
zero speed
according to some positive parameter $\alpha >1$ or $\le 1$. In this
article,
we give the exact rate of growth of $(X_n)$ in the zero speed regime,
namely:
for $0<\alpha <1$, $X_n/n^{\frac{\alpha+1}{2}}$ converges in law to a
Mittag-Leffler distribution whereas for $\alpha=1$, $X_n(\log n)/n$
converges
in probability to some positive constant.
http://arXiv.org/abs/math/0703275
---------------------------------------------------------------
5316. TRANSITION BETWEEN AIRY_1 AND AIRY_2 PROCESSES AND TASEP
FLUCTUATIONS
Alexei Borodin (1) and Patrik L. Ferrari (2) and Tomohiro Sasamoto
(3) ((1) Caltech, (2) WIAS Berlin, (3) Chiba University)
We consider the totally asymmetric simple exclusion process, a model
in the
KPZ universality class. We focus on the fluctuations of particle
positions
starting with certain deterministic initial conditions. For large
time t, one
has regions with constant and linearly decreasing density. The
fluctuations on
these two regions are given by the Airy_1 and Airy_2 processes, whose
one-point
distributions are the GOE and GUE Tracy-Widom distributions of random
matrix
theory. In this paper we analyze the transition region between these two
regimes and obtain the transition process. Its one-point distribution
is a new
interpolation between GOE and GUE edge distributions.
http://arXiv.org/abs/math-ph/0703023
---------------------------------------------------------------
5317. PATH INTEGRALS ON MANIFOLDS BY FINITE DIMENSIONAL APPROXIMATION
Christian Baer and Frank Pfaeffle
Let M be a compact Riemannian manifold without boundary and let H be a
self-adjoint generalized Laplace operator acting on sections in a
bundle over
M. We give a path integral formula for the solution to the
corresponding heat
equation. This is based on approximating path space by finite
dimensional
spaces of geodesic polygons. We also show a uniform convergence
result for the
heat kernels. This yields a simple and natural proof for the
Hess-Schrader-Uhlenbrock estimate and a path integral formula for the
trace of
the heat operator.
http://arXiv.org/abs/math/0703272
---------------------------------------------------------------
5318. PERCOLATION ON SPARSE RANDOM GRAPHS WITH GIVEN DEGREE SEQUENCE
Nikolaos Fountoulakis
We study the two most common types of percolation process on a sparse
random
graph with a given degree sequence. Namely, we examine first a bond
percolation
process where the edges of the graph are retained with probability p and
afterwards we focus on site percolation where the vertices are
retained with
probability p. We establish critical values for p above which a giant
component
emerges in both cases. Moreover, we show that in fact these coincide.
As a
special case, our results apply to power law random graphs. We obtain
rigorous
proofs for formulas derived by several physicists for such graphs.
http://arXiv.org/abs/math/0703269
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5319. EXISTENCE AND UNIQUENESS OF NONNEGATIVE SOLUTIONS TO THE
STOCHASTIC POROUS MEDIA EQUATION
Viorel Barbu and Giuseppe Da Prato and Michael R\"ockner
One proves that the stochastic porous media equation in 3-D has a unique
nonnegative solution for nonnegative initial data in $H^{-1}(\mathcal
O)$ if
the nonlinearity is monotone and has polynomial growth.
http://arXiv.org/abs/math/0703420
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5320. EXISTENCE OF STRONG SOLUTIONS FOR STOCHASTIC POROUS MEDIA
EQUATION UNDER GENERAL MONOTONICITY CONDITIONS
Viorel Barbu and Giuseppe Da Prato and Michael R\"ockner
One proves existence and uniqueness of strong solutions to stochastic
porous
media equations under minimal monotonicity conditions on the
nonlinearity. In
particular, we do not assume continuity of the drift or any growth
condition at
infinity.
http://arXiv.org/abs/math/0703421
---------------------------------------------------------------
5321. QUADRATIC BSDES WITH CONVEX GENERATORS AND UNBOUNDED TERMINAL
CONDITIONS
Philippe Briand (IRMAR) and Ying Hu (IRMAR)
In a previous work, we proved an existence result for BSDEs with
quadratic
generators with respect to the variable z and with unbounded terminal
conditions. However, no uniqueness result was stated in that work.
The main
goal of this paper is to fill this gap. In order to obtain a
comparison theorem
for this kind of BSDEs, we assume that the generator is convex with
respect to
the variable z. Under this assumption of convexity, we are also able
to prove a
stability result in the spirit of the a priori estimates stated in
the article
of N. El Karoui, S. Peng and M.-C. Quenez. With these tools in hands,
we can
derive the nonlinear Feynman--Kac formula in this context.
http://arXiv.org/abs/math/0703423
---------------------------------------------------------------
5322. MEAN-VARIANCE HEDGING UNDER PARTIAL INFORMATION
M. Mania and R. Tevzadze and T. Toronjadze
We consider the mean-variance hedging problem under partial
Information. The
underlying asset price process follows a continuous semimartingale and
strategies have to be constructed when only part of the information
in the
market is available. We show that the initial mean variance hedging
problem is
equivalent to a new mean variance hedging problem with an additional
correction
term, which is formulated in terms of observable processes. We prove
that the
value process of the reduced problem is a square trinomial with
coefficients
satisfying a triangle system of backward stochastic differential
equations and
the filtered wealth process of the optimal hedging strategy is
characterized as
a solution of a linear forward equation.
http://arXiv.org/abs/math/0703424
---------------------------------------------------------------
5323. MEASURABILITY OF OPTIMAL TRANSPORTATION AND CONVERGENCE RATE
FOR LANDAU TYPE INTERACTING PARTICLE SYSTEMS
Joaquin Fontbona and Helene Guerin and Sylvie Meleard
In this paper, we consider nonlinear diffusion processes driven by
space-time
white noises, which have an interpretation in terms of partial
differential
equations. For a specific choice of coefficients, they correspond to
the Landau
equation arising in kinetic theory. A particular feature is that the
diffusion
matrix of this process is a linear function the law of the process,
and not a
quadratic one, as in the McKean-Vlasov model. The main goal of the
paper is to
construct an easily simulable diffusive interacting particle system,
converging
towards this nonlinear process and to obtain an explicit pathwise rate.
This requires to find a significant coupling between finitely many
Brownian
motions and the infinite dimensional white noise process. The key
idea will be
to construct the right Brownian motions by pushing forward the white
noise
processes, through the Brenier map realizing the optimal transport
between the
law of the nonlinear process, and the empirical measure of
independent copies
of it. A striking problem then is to establish the joint
measurability of this
optimal transport map with respect to the space variable and the
parameters
(time and randomness) making the marginals vary. We shall prove a
general
measurability result for the mass transportation problem in terms of the
support of the transfert plans, in the sense of set-valued mappings.
This will
allow us to construct the coupling and to obtain explicit convergence
rates.
http://arXiv.org/abs/math/0703432
---------------------------------------------------------------
5324. ON A MODEL OF RANDOM CYCLES
Daniel Gandolfo and Jean Ruiz and Daniel Ueltschi
We introduce a model of random permutations of the sites of the cubic
lattice. Permutations are weighted so that sites are preferably sent
onto
neighbors. We present numerical evidence for the occurrence of a
transition to
a phase with infinite, macroscopic cycles.
http://arXiv.org/abs/cond-mat/0703315
---------------------------------------------------------------
5325. THE SMALL DEVIATIONS OF MANY-DIMENSIONAL DIFFUSION PROCESSES
AND RAREFACTION BY BOUNDARIES
Vitalii A. Gasanenko
We lead the algorithm of expansion of sojourn probability of many-
dimensional
diffusion processes in small domain. The principal member of this
expansion
defines normalizing coefficient for special limit theorems.
http://arxiv.org/abs/0704.0315
---------------------------------------------------------------
5326. SOLUTIONS OF FRACTIONAL REACTION-DIFFUSION EQUATIONS IN TERMS
OF THE H-FUNCTION
H.J. Haubold and A.M. Mathai and R.K. Saxena
This paper deals with the investigation of the solution of an unified
fractional reaction-diffusion equation associated with the Caputo
derivative as
the time-derivative and Riesz-Feller fractional derivative as the
space-derivative. The solution is derived by the application of the
Laplace and
Fourier transforms in closed form in terms of the H-function. The
results
derived are of general nature and include the results investigated
earlier by
many authors, notably by Mainardi et al. (2001, 2005) for the
fundamental
solution of the space-time fractional diffusion equation, and Saxena
et al.
(2006a, b) for fractional reaction- diffusion equations. The
advantage of using
Riesz-Feller derivative lies in the fact that the solution of the
fractional
reaction-diffusion equation containing this derivative includes the
fundamental
solution for space-time fractional diffusion, which itself is a
generalization
of neutral fractional diffusion, space-fractional diffusion, and
time-fractional diffusion. These specialized types of diffusion can be
interpreted as spatial probability density functions evolving in time
and are
expressible in terms of the H-functions in compact form.
http://arxiv.org/abs/0704.0329
---------------------------------------------------------------
5327. APPROXIMATION OF THE DISTRIBUTION OF A STATIONARY MARKOV
PROCESS WITH APPLICATION TO OPTION PRICING
Fabien Panloup (PMA) and Gilles Pag{\`e}s (PMA)
We build a sequence of empirical measures on the space D(R_+,R^d) of
R^d-valued c{\`a}dl{\`a}g functions on R_+ in order to approximate
the law of a
stationary R^d-valued Markov and Feller process (X_t). We obtain some
general
results of convergence of this sequence. Then, we apply them to Brownian
diffusions and solutions to L{\'e}vy driven SDE's under some Lyapunov-
type
stability assumptions. As a numerical application of this work, we
show that
this procedure gives an efficient way of option pricing in stochastic
volatility models.
http://arxiv.org/abs/0704.0335
---------------------------------------------------------------
5328. EXPONENTIAL GROWTH RATES IN A TYPED BRANCHING DIFFUSION
Y. Git and J. W. Harris and S. C. Harris
We study the high temperature phase of a family of typed branching
diffusions
initially studied in [Ast\'{e}risque 236 (1996) 133--154] and
[Lecture Notes in
Math. 1729 (2000) 239--256 Springer, Berlin]. The primary aim is to
establish
some almost-sure limit results for the long-term behavior of this
particle
system, namely the speed at which the population of particles
colonizes both
space and type dimensions, as well as the rate at which the
population grows
within this asymptotic shape. Our approach will include
identification of an
explicit two-phase mechanism by which particles can build up in
sufficient
numbers with spatial positions near $-\gamma t$ and type positions
near $\kappa
\sqrt{t}$ at large times $t$. The proofs involve the application of a
variety
of martingale techniques--most importantly a ``spine'' construction
involving a
change of measure with an additive martingale. In addition to the
model's
intrinsic interest, the methodologies presented contain ideas that
will adapt
to other branching settings. We also briefly discuss applications to
traveling
wave solutions of an associated reaction--diffusion equation.
http://arxiv.org/abs/0704.0380
---------------------------------------------------------------
5329. AVERAGE OPTIMALITY FOR RISK-SENSITIVE CONTROL WITH GENERAL
STATE SPACE
Anna Ja\'{s}kiewicz
This paper deals with discrete-time Markov control processes on a
general
state space. A long-run risk-sensitive average cost criterion is used
as a
performance measure. The one-step cost function is nonnegative and
possibly
unbounded. Using the vanishing discount factor approach, the optimality
inequality and an optimal stationary strategy for the decision maker are
established.
http://arxiv.org/abs/0704.0394
---------------------------------------------------------------
5330. RENEWALS FOR EXPONENTIALLY INCREASING LIFETIMES, WITH AN
APPLICATION TO DIGITAL SEARCH TREES
Florian Dennert and Rudolf Gr\"{u}bel
We show that the number of renewals up to time $t$ exhibits
distributional
fluctuations as $t\to\infty$ if the underlying lifetimes increase at an
exponential rate in a distributional sense. This provides a
probabilistic
explanation for the asymptotics of insertion depth in random trees
generated by
a bit-comparison strategy from uniform input; we also obtain a
representation
for the resulting family of limit laws along subsequences. Our
approach can
also be used to obtain rates of convergence.
http://arxiv.org/abs/0704.0398
---------------------------------------------------------------
5331. AN INVARIANCE PRINCIPLE FOR SEMIMARTINGALE REFLECTING BROWNIAN
MOTIONS IN DOMAINS WITH PIECEWISE SMOOTH BOUNDARIES
W. Kang and R. J. Williams
Semimartingale reflecting Brownian motions (SRBMs) living in the
closures of
domains with piecewise smooth boundaries are of interest in applied
probability
because of their role as heavy traffic approximations for some
stochastic
networks. In this paper, assuming certain conditions on the domains and
directions of reflection, a perturbation result, or invariance
principle, for
SRBMs is proved. This provides sufficient conditions for a process that
satisfies the definition of an SRBM, except for small random
perturbations in
the defining conditions, to be close in distribution to an SRBM. A
crucial
ingredient in the proof of this result is an oscillation inequality for
solutions of a perturbed Skorokhod problem. We use the invariance
principle to
show weak existence of SRBMs under mild conditions. We also use the
invariance
principle, in conjunction with known uniqueness results for SRBMs, to
give some
sufficient conditions for validating approximations involving (i)
SRBMs in
convex polyhedrons with a constant reflection vector field on each
face of the
polyhedron, and (ii) SRBMs in bounded domains with piecewise smooth
boundaries
and possibly nonconstant reflection vector fields on the boundary
surfaces.
http://arxiv.org/abs/0704.0405
---------------------------------------------------------------
5332. SOLUTIONS OF FRACTIONAL REACTION-DIFFUSION EQUATIONS IN TERMS
OF THE H-FUNCTION
H.J. Haubold and A.M. Mathai and R.K. Saxena
This paper deals with the investigation of the solution of an unified
fractional reaction-diffusion equation associated with the Caputo
derivative as
the time-derivative and Riesz-Feller fractional derivative as the
space-derivative. The solution is derived by the application of the
Laplace and
Fourier transforms in closed form in terms of the H-function. The
results
derived are of general nature and include the results investigated
earlier by
many authors, notably by Mainardi et al. (2001, 2005) for the
fundamental
solution of the space-time fractional diffusion equation, and Saxena
et al.
(2006a, b) for fractional reaction- diffusion equations. The
advantage of using
Riesz-Feller derivative lies in the fact that the solution of the
fractional
reaction-diffusion equation containing this derivative includes the
fundamental
solution for space-time fractional diffusion, which itself is a
generalization
of neutral fractional diffusion, space-fractional diffusion, and
time-fractional diffusion. These specialized types of diffusion can be
interpreted as spatial probability density functions evolving in time
and are
expressible in terms of the H-functions in compact form.
http://arxiv.org/abs/0704.0329
---------------------------------------------------------------
5333. QUENCHED LIMITS FOR TRANSIENT, ZERO SPEED ONE-DIMENSIONAL
RANDOM WALK IN RANDOM ENVIRONMENT
Jonathon Peterson and Ofer Zeitouni
We consider a nearest-neighbor, one dimensional random walk $\{X_n\}_
{n\geq
0}$ in a random i.i.d. environment, in the regime where the walk is
transient
but with zero speed, so that $X_n$ is of order $n^{s}$ for some $s<1
$. Under
the quenched law (i.e., conditioned on the environment), we show that
no limit
laws are possible: there exist sequences $\{n_k\}$ and $\{x_k\}$
depending on
the environment only, such that $X_{n_k}-x_k=o(\log n_k)^2$ (a localized
regime). On the other hand, there exist sequences $\{t_m\}$ and $\{s_m
\}$
depending on the environment only, such that $\log t_m/\log s_m\to s<1
$ and
$P_\omega(X_{t_m}/s_m\leq x)\to 1/2$ for all $x>0$ and $\to 0$ for $x
\leq 0$ (a
spread out regime).
http://arxiv.org/abs/0704.1778
---------------------------------------------------------------
5334. REPRESENTATION THEOREMS FOR QUADRATIC ${\CAL F}$-CONSISTENT
NONLINEAR EXPECTATIONS
Ying Hu (IRMAR) and Jin Ma (Department of Mathematics) and Shige
Peng (Institute of Mathematics), Song Yao (Department of Mathematics)
In this paper we extend the notion of ``filtration-consistent nonlinear
expectation" (or "${\cal F}$-consistent nonlinear expectation") to
the case
when it is allowed to be dominated by a $g$-expectation that may have a
quadratic growth. We show that for such a nonlinear expectation many
fundamental properties of a martingale can still make sense,
including the
Doob-Meyer type decomposition theorem and the optional sampling
theorem. More
importantly, we show that any quadratic ${\cal F}$-consistent nonlinear
expectation with a certain domination property must be a quadratic
$g$-expectation. The main contribution of this paper is the finding
of the
domination condition to replace the one used in all the previous
works, which
is no longer valid in the quadratic case. We also show that the
representation
generator must be deterministic, continuous, and actually must be of
the simple
form.
http://arxiv.org/abs/0704.1796
---------------------------------------------------------------
5335. GENERALIZED SMIRNOV STATISTICS AND THE DISTRIBUTION OF PRIME
FACTORS
Kevin Ford
We apply recent bounds of the author (math.PR/0609224) for generalized
Smirnov statistics to the distribution of integers whose prime
factors satisfy
certain systems of inequalities.
http://arxiv.org/abs/0704.1789
---------------------------------------------------------------
5336. TYPICAL SUPPORT AND SANOV LARGE DEVIATIONS OF CORRELATED STATES
I. Bjelakovic and J.-D. Deuschel and T. Krueger and R. Seiler and
Ra. Siegmund-Schultze, A. Szkola
Discrete stationary classical processes as well as quantum lattice
states are
asymptotically confined to their respective typical support, the
exponential
growth rate of which is given by the (maximal ergodic) entropy. In
the iid case
the distinguishability of typical supports can be asymptotically
specified by
means of the relative entropy, according to Sanov's theorem. We give an
extension to the correlated case, referring to the newly introduced
class of
HP-states.
http://arXiv.org/abs/math/0703772
---------------------------------------------------------------
5337. QUASI-STATIONARITY FOR POPULATION DIFFUSION PROCESSES
Patrick Cattiaux (CMAP and LSProba) and Pierre Collet (CPHT) and
Amaury Lambert (FESE), Servet Martinez (CMM), Sylvie M{\'e}l{\'e}ard
(CMAP), Jaime San
Martin (CMM)
In this paper, we study quasi-stationarity for a large class of
Kolmogorov
diffusions, that is, existence of a quasi-stationary distribution,
conditional
convergence to such a distribution, construction of a $Q$-process
(process
conditioned to be never extinct). The main novelty here is that we
allow the
drift to go to $- \infty$ at the origin, and the diffusion to have an
entrance
boundary at $+\infty$. These diffusions arise as images, by a
deterministic
map, of generalized Feller diffusions, which themselves are obtained
as limits
of rescaled birth--death processes. Generalized Feller diffusions take
non-negative values and are absorbed at zero in finite time with
probability 1.
A toy example is the logistic Feller diffusion. We give sufficient
conditions
on the drift near 0 and near $+ \infty$ for the existence of quasi-
stationary
distributions, as well as rate of convergence, and existence of the
$Q$-process. We also show that under these conditions, there is
exactly one
conditional limiting distribution (which implies uniqueness of the
quasi-stationary distribution) if and only if the process comes down
from
infinity. Proofs are based on spectral theory. Here the reference
measure is
the natural symmetric measure for the killed process, and we use in an
essential way the Girsanov transform.
http://arXiv.org/abs/math/0703781
---------------------------------------------------------------
5338. QUENCHED INVARIANCE PRINCIPLE FOR MULTIDIMENSIONAL BALLISTIC
RANDOM WALK IN A RANDOM ENVIRONMENT WITH A FORBIDDEN DIRECTION
Firas Rassoul-Agha and Timo Sepp\"{a}l\"{a}inen
We consider a ballistic random walk in an i.i.d. random environment
that does
not allow retreating in a certain fixed direction. We prove an
invariance
principle (functional central limit theorem) under almost every fixed
environment. The assumptions are nonnestling, at least two spatial
dimensions,
and a $2+\epsilon$ moment for the step of the walk uniformly in the
environment. The main point behind the invariance principle is that the
quenched mean of the walk behaves subdiffusively.
http://arXiv.org/abs/math/0703787
---------------------------------------------------------------
5339. UN TH\'{E}OR\`{E}ME LIMITE POUR LES COVARIANCES DES SPINS DANS
LE MOD\`{E}LE DE SHERRINGTON--KIRKPATRICK AVEC CHAMP EXTERNE
Albert Hanen
On \'{e}tudie la covariance (pour la mesure de Gibbs) des spins en
deux sites
dans le cas d'un mod\`{e}le de Sherrington--Kirkpatrick avec champ
externe;
lorsque le nombre de sites du mod\`{e}le tend vers l'infini, une \'{e}
valuation
asymptotique des moments d'ordre $p$ de cette covariance permet
d'obtenir un
th\'{e}or\`{e}me limite faible avec une loi limite en g\'{e}n\'{e}ral
non
gaussienne. We study the covariance (for Gibbs measure) of spins at
two sites
in the case of a Sherrington--Kirkpatrick model with an external
field. When
the number of sites of the model grows to infinity, an asymptotic
evaluation of
the $p$ moments of that covariance allows us to obtain a weak limit
theorem,
with a generally non-Gaussian limit law.
http://arXiv.org/abs/math/0703790
---------------------------------------------------------------
5340. GLOBAL FLOWS FOR STOCHASTIC DIFFERENTIAL EQUATIONS WITHOUT
GLOBAL LIPSCHITZ CONDITIONS
Shizan Fang and Peter Imkeller and Tusheng Zhang
We consider stochastic differential equations driven by Wiener
processes. The
vector fields are supposed to satisfy only local Lipschitz
conditions. The
Lipschitz constants of the drift vector field, valid on balls of
radius $R$,
are supposed to grow not faster than $\log R$, while those of the
diffusion
vector fields are supposed to grow not faster than $\sqrt{\log R}.$ We
regularize the stochastic differential equations by associating with
them
approximating ordinary differential equations obtained by
discretization of the
increments of the Wiener process on small intervals. By showing that
the flow
associated with a regularized equation converges uniformly to the
solution of
the stochastic differential equation, we simultaneously establish the
existence
of a global flow for the stochastic equation under local Lipschitz
conditions.
http://arXiv.org/abs/math/0703791
---------------------------------------------------------------
5341. COMPARISON OF SEMIMARTINGALES AND L\'{E}VY PROCESSES
Jan Bergenthum and Ludger R\"{u}schendorf
In this paper, we derive comparison results for terminal values of
$d$-dimensional special semimartingales and also for finite-dimensional
distributions of multivariate L\'{e}vy processes. The comparison is with
respect to nondecreasing, (increasing) convex, (increasing)
directionally
convex and (increasing) supermodular functions. We use three different
approaches. In the first approach, we give sufficient conditions on
the local
predictable characteristics that imply ordering of terminal values of
semimartingales. This generalizes some recent convex comparison
results of
exponential models in [Math. Finance 8 (1998) 93--126, Finance Stoch.
4 (2000)
209--222, Proc. Steklov Inst. Math. 237 (2002) 73--113, Finance
Stoch. 10
(2006) 222--249]. In the second part, we give comparison results for
finite-dimensional distributions of L\'{e}vy processes with infinite L
\'{e}vy
measure. In the first step, we derive a comparison result for Markov
processes
based on a monotone separating transition kernel. By a coupling
argument, we
get an application to the comparison of compound Poisson processes.
These
comparisons are then extended by an approximation argument to the
ordering of
L\'{e}vy processes with infinite L\'{e}vy measure. The third approach
is based
on mixing representations which are known for several relevant
distribution
classes. We discuss this approach in detail for the comparison of
generalized
hyperbolic distributions and for normal inverse Gaussian processes.
http://arXiv.org/abs/math/0703793
---------------------------------------------------------------
5342. ASYMPTOTIC DEVELOPMENTS AT ANY TIME FOR FRACTIONAL SDES OF
HURST INDEX H>1/2
S\'ebastien Darses (LM-Besan\c{c}on) and Ivan Nourdin (LM-Besan\c{c}on)
We study the asymptotic developments with respect to $h$ of E[D_h f
(X_t)],
E[D_h f(X_t)|F_t] and E[D_h f(X_t)|X_t], where D_h f(X_t)=f(X_{t+h})-f
(X_t),
when f:R->R is a smooth real function, t is a fixed time, X is the
solution of
a one-dimensional stochastic differential equation driven by a
fractional
Brownian motion of Hurst index H>1/2 and F is its natural filtration.
http://arXiv.org/abs/math/0703794
---------------------------------------------------------------
5343. EXTREMAL BEHAVIOR OF STOCHASTIC INTEGRALS DRIVEN BY REGULARLY
VARYING L\'{E}VY PROCESSES
Henrik Hult and Filip Lindskog
We study the extremal behavior of a stochastic integral driven by a
multivariate L\'{e}vy process that is regularly varying with index $
\alpha>0$.
For predictable integrands with a finite $(\alpha+\delta)$-moment,
for some
$\delta>0$, we show that the extremal behavior of the stochastic
integral is
due to one big jump of the driving L\'{e}vy process and we determine
its limit
measure associated with regular variation on the space of c\`{a}dl\`{a}g
functions.
http://arXiv.org/abs/math/0703802
---------------------------------------------------------------
5344. THE TRAP OF COMPLACENCY IN PREDICTING THE MAXIMUM
J. du Toit and G. Peskir
Given a standard Brownian motion $B^{\mu}=(B_t^{\mu})_{0\le t\le T}$
with
drift $\mu \in \mathbb{R}$ and letting $S_t^{\mu}=\max_{0\le s\le t}
B_s^{\mu}$
for $0\le t\le T$, we consider the optimal prediction problem: \[V=
\inf_{0\le
\tau \le T}\mathsf{E}(B_{\tau}^{\mu}-S_T^{\mu})^2\] where the infimum
is taken
over all stopping times $\tau$ of $B^{\mu}$. Reducing the optimal
prediction
problem to a parabolic free-boundary problem we show that the following
stopping time is optimal: \[\tau_*=\inf \{t_*\le t\le T\mid b_1(t)\le
S_t^{\mu}-B_t^{\mu}\le b_2(t)\}\] where $t_*\in [0,T)$ and the functions
$t\mapsto b_1(t)$ and $t\mapsto b_2(t)$ are continuous on $[t_*,T]$ with
$b_1(T)=0$ and $b_2(T)=1/2\mu$. If $\mu>0$, then $b_1$ is decreasing
and $b_2$
is increasing on $[t_*,T]$ with $b_1(t_*)=b_2(t_*)$ when $t_*\ne 0$.
Using
local time-space calculus we derive a coupled system of nonlinear
Volterra
integral equations of the second kind and show that the pair of optimal
boundaries $b_1$ and $b_2$ can be characterized as the unique
solution to this
system. This also leads to an explicit formula for $V$ in terms of
$b_1$ and
$b_2$. If $\mu \le 0$, then $t_*=0$ and $b_2\equiv +\infty$ so that $
\tau_*$ is
expressed in terms of $b_1$ only. In this case $b_1$ is decreasing on
$[z_*,T]$
and increasing on $[0,z_*)$ for some $z_*\in [0,T)$ with $z_*=0$ if $
\mu=0$,
and the system of two Volterra equations reduces to one Volterra
equation. If
$\mu=0$, then there is a closed form expression for $b_1$. This
problem was
solved in [Theory Probab. Appl. 45 (2001) 125--136] using the method
of time
change (i.e., change of variables). The method of time change cannot be
extended to the case when $\mu \ne 0$ and the present paper settles the
remaining cases using a different approach.
http://arXiv.org/abs/math/0703805
---------------------------------------------------------------
5345. MULTIVARIABLE APPROXIMATE CARLEMAN-TYPE THEOREMS FOR COMPLEX
MEASURES
Isabelle Chalendar and Jonathan R. Partington
We prove a multivariable approximate Carleman theorem on the
determination of
complex measures on ${\mathbb{R}}^n$ and ${\mathbb{R}}^n_+$ by their
moments.
This is achieved by means of a multivariable Denjoy--Carleman maximum
principle
for quasi-analytic functions of several variables. As an application,
we obtain
a discrete Phragm\'{e}n--Lindel\"{o}f-type theorem for analytic
functions on
${\mathbb{C}}_+^n$.
http://arXiv.org/abs/math/0703809
---------------------------------------------------------------
5346. A PROOF OF THE SMOOTHNESS OF THE FINITE TIME HORIZON AMERICAN
PUT OPTION FOR JUMP DIFFUSIONS
Erhan Bayraktar
We give a new proof of the fact that the value function of the finite
time
horizon American put option for a jump diffusion, when the jumps are
from a
compound Poisson process, is the classical solution of a quasi-
variational
inequality and it is $C^1$ across the optimal stopping boundary. Our
proof only
uses the classical theory of parabolic partial differential equations of
\cite{friedmansde} and does not use the \emph{the theory of vicosity
solutions}, since our proof relies on constructing a sequence of
functions,
each of which is a value function of an optimal stopping time for a
\emph{diffusion}. The sequence is constructed by iterating a functional
operator that maps a certain class of convex functions to smooth
functions
satisfying variational inequalities (or to value functions of optimal
stopping
problems involving only a diffusion). The approximating sequence
converges to
the value function exponentially fast, therefore it constitutes a good
approximation scheme, since the optimal stopping problems for
diffusions can be
readily solved. Our technique also lets one see why the jump-
diffusion control
problems may be smoother than the control problems with piece-wise
deterministic Markov processes: In the former case the sequence of
functions
that converge to the value function is a sequence of value function
of control
problems for diffusions, and in the latter case the converging
sequence is a
sequence of the value functions of deterministic optimal control
problems. The
first of these sequences is known to be smoother than the second one.
http://arXiv.org/abs/math/0703782
---------------------------------------------------------------
5347. EXISTENCE AND STABILITY FOR FOKKER-PLANCK EQUATIONS WITH LOG-
CONCAVE REFERENCE MEASURE
Luigi Ambrosio and Giuseppe Savare and Lorenzo Zambotti
We study Markov processes associated with stochastic differential
equations,
whose non-linearities are gradients of convex functionals. We prove a
general
result of existence of such Markov processes and a priori estimates
on the
transition probabilities. The main result is the following stability
property:
if the associated invariant measures converge weakly, then the Markov
processes
converge in law. The proofs are based on the interpretation of a
Fokker-Planck
equation as the steepest descent flow of the relative Entropy
functional in the
space of probability measures, endowed with the Wasserstein distance.
Applications include stochastic partial differential equations and
convergence
of equilibrium fluctuations for a class of random interfaces.
http://arxiv.org/abs/0704.2458
---------------------------------------------------------------
5348. VACANT SET OF RANDOM INTERLACEMENTS AND PERCOLATION
Alain-Sol Sznitman
We introduce a model of random interlacements made of a countable
collection
of doubly infinite paths on Z^d, d bigger or equal to 3. A non-negative
parameter u measures how many trajectories enter the picture. This model
describes in the large N limit the microscopic structure in the bulk,
which
arises when considering the disconnection time of a discrete cylinder
with base
a d-1 dimensional discrete torus of side-length N, or the set of
points visited
by simple random walk on the d dimensional discrete torus of side-
length N by
times of order uN^d. We study the percolative properties of the
vacant set left
by the interlacement at level u, which is an infinite connected
translation
invariant random subset of Z^d. We introduce a critical value such
that the
vacant set percolates for u below the critical value, and does not
percolate
for u above the critical value. Our main results show that the
critical value
is finite when d is bigger or equal to 3, and strictly positive when
d is
bigger or equal to 7.
http://arxiv.org/abs/0704.2560
---------------------------------------------------------------
5349. DISTRIBUTIONS OF ROOTS OF REDUCED CUBIC EQUATIONS WITH RANDOM
COEFFICIENTS
Kerry M. Soileau
If the coefficients of polynomials are selected by some random
process, the
zeros of the resulting polynomials are in some sense random. In this
paper the
author rephrases the above in more precise language, and calculates
the joint
conditional densities of a random vector whose values determine
almost surely
the zeros of a "random" reduced cubic.
http://arxiv.org/abs/0704.2586
---------------------------------------------------------------
5350. STRUCTURAL ADAPTATION VIA $L_P$-NORM ORACLE INEQUALITIES
A. Goldenhsluger and O. Lepski
In this paper we study the problem of adaptive estimation of a
multivariate
function satisfying some structural assumption. We propose a novel
estimation
procedure that adapts simultaneously to unknown structure and
smoothness of the
underlying function. The problem of structural adaptation is stated
as the
problem of selection from a given collection of estimators. We develop a
general selection rule and establish for it global oracle
inequalities under
arbitrary $\rL_p$--losses. These results are applied for adaptive
estimation in
the additive multi--index model.
http://arxiv.org/abs/0704.2492
---------------------------------------------------------------
5351. A QUENCHED CLT FOR SUPER-BROWNIAN MOTION WITH RANDOM IMMIGRATION
Wenming Hong and Ofer Zeitouni
A quenched central limit theorem is derived for the super-Brownian
motion
with super-Brownian immigration, in dimension $d\geq 4$. At the critical
dimension $d=4$, the quenched and annealed fluctuations are of the
same order
but are not equal.
http://arXiv.org/abs/math/0703573
---------------------------------------------------------------
5352. ON AN EXPLICIT SKOROKHOD EMBEDDING FOR SPECTRALLY NEGATIVE
LEVY PROCESSES
Jan Obloj and Martijn Pistorius
We solve explicitly the Skorokhod embedding problem for spectrally
negative
L\'evy processes. Given a process $X$ and a target measure $\mu$
satisfying
explicit admissibility condition we provide functions $\f_\pm$ such
that the
stopping time $T = \inf\{t>0: X_t \in \{-\f_-(L_t), \f_+(L_t)\}\}$
induces
$X_T\sim \mu$. We also treat versions of $T$ which take into account
the sign
of the excursion straddling time $t$. We prove that our stopping
times are
minimal and we describe criteria under which they are integrable. Our
method
relies on some new explicit calculations relating scale functions and
the It\^o
excursion measure of $X$. Finally, we compare our solution with the one
proposed by Bertoin and Le Jan (1992). In particular, we compute
explicitly
their general quantities in our setup.
http://arXiv.org/abs/math/0703597
---------------------------------------------------------------
5353. USE OF AN HOURGLASS MODEL IN NEURONAL CODING
Marie Cottrell (SAMOS and Matisse) and Tatiana Turova (DMS Lund)
We study a system of interacting renewal processes which is a model for
neuronal activity. We show that the system possesses an exponentially
large
number (with respect to the number of neurons in the network) of
limiting
configurations of the "firing neurons". These we call patterns.
Furthermore,
under certain conditions of symmetry we find an algorithm to control
limiting
patterns by means of the connection parameters.
http://arXiv.org/abs/math/0703010
---------------------------------------------------------------
5354. ASYMPTOTIC DISTRIBUTIONS OF THE SIGNAL-TO-INTERFERENCE RATIOS
OF LMMSE DETECTION IN MULTIUSER COMMUNICATIONS
Guang-Ming Pan and Mei-Hui Guo and Wang Zhou
Let ${\mathbf{s}}_k=\frac{1}{\sqrt{N}}(v_{1k},...,v_{Nk})^T,$
$k=1,...,K$,
where $\{v_{ik},i,k$ $=1,...\}$ are independent and identically
distributed
random variables with $Ev_{11}=0$ and $Ev_{11}^2=1$. Let
${\mathbf{S}}_k=({\mathbf{s}}_1,...,{\mathbf{s}}_{k-1},$
${\mathbf{s}}_{k+1},...,{\mathbf{s}}_K)$, ${\mathbf{P}}_k=\operatorname
{diag}(p_1,...,$ $p_{k-1},p_{k+1},...,p_K)$ and
$\beta_k=p_k{\mathbf{s}}_k^T({\mathb
f{S}}_k{\mathbf{P}}_k{\mathbf{S}}_k^T+\sigma^2{\mathbf{I}})^{-1}{\math
bf{s}}_k$, where $p_k\geq 0$ and the $\beta_k$ is referred to as the
signal-to-interference ratio (SIR) of user $k$ with linear minimum
mean-square
error (LMMSE) detection in wireless communications. The joint
distribution of
the SIRs for a finite number of users and the empirical distribution
of all
users' SIRs are both investigated in this paper when $K$ and $N$ tend to
infinity with the limit of their ratio being positive constant.
Moreover, the
sum of the SIRs of all users, after subtracting a proper value, is
shown to
have a Gaussian limit.
http://arXiv.org/abs/math/0703014
---------------------------------------------------------------
5355. SINGULARLY PERTURBED MARKOV CHAINS: LIMIT RESULTS AND APPLICATIONS
George Yin and Hanqin Zhang
This work focuses on time-inhomogeneous Markov chains with two time
scales.
Our motivations stem from applications in reliability and dependability,
queueing networks, financial engineering and manufacturing systems,
where
two-time-scale scenarios naturally arise. One of the important
questions is: As
the rate of fluctuation of the Markov chain goes to infinity, if the
limit
distributions of suitably centered and scaled sequences of occupation
measures
exist, what can be said about the convergence rate? By combining
singular
perturbation techniques and probabilistic methods, this paper
addresses the
issue by concentrating on sequences of centered and scaled functional
occupation processes. The results obtained are then applied to treat
a queueing
system example.
http://arXiv.org/abs/math/0703017
---------------------------------------------------------------
5356. POISSON LIMITS OF SUMS OF POINT PROCESSES AND A PARTICLE-
SURVIVOR MODEL
Matthew O. Jones and Richard F. Serfozo
We present sufficient conditions for sums of dependent point
processes to
converge in distribution to a Poisson process. This extends the
classical
result of Grigelionis [Theory Probab. Appl. 8 (1963) 172--182] for
sums of
uniformly null point processes that have Poisson limits. Included is an
application in which a particle-survivor point process converges to a
Poisson
process. This result sheds light on the ``surprising'' Poisson limit
of the
species competition process of Durrett and Limic [Stochastic Process.
Appl. 102
(2002) 301--309].
http://arXiv.org/abs/math/0703018
---------------------------------------------------------------
5357. READING POLICIES FOR JOINS: AN ASYMPTOTIC ANALYSIS
Ralph P. Russo and Nariankadu D. Shyamalkumar
Suppose that $m_n$ observations are made from the distribution $
\mathbf {R}$
and $n-m_n$ from the distribution $\mathbf {S}$. Associate with each
pair, $x$
from $\mathbf {R}$ and $y$ from $\mathbf {S}$, a nonnegative score $
\phi(x,y)$.
An optimal reading policy is one that yields a sequence $m_n$ that
maximizes
$\mathbb{E}(M(n))$, the expected sum of the $(n-m_n)m_n$ observed
scores,
uniformly in $n$. The alternating policy, which switches between the two
sources, is the optimal nonadaptive policy. In contrast, the greedy
policy,
which chooses its source to maximize the expected gain on the next
step, is
shown to be the optimal policy. Asymptotics are provided for the case
where the
$\mathbf {R}$ and $\mathbf {S}$ distributions are discrete and $\phi
(x,y)=1 or
0$ according as $x=y$ or not (i.e., the observations match).
Specifically, an
invariance result is proved which guarantees that for a wide class of
policies,
including the alternating and the greedy, the variable M(n) obeys the
same CLT
and LIL. A more delicate analysis of the sequence $\mathbb{E}(M(n))$
and the
sample paths of M(n), for both alternating and greedy, reveals the
slender
sense in which the latter policy is asymptotically superior to the
former, as
well as a sense of equivalence of the two and robustness of the former.
http://arXiv.org/abs/math/0703019
---------------------------------------------------------------
5358. SMALL-WORLD MCMC AND CONVERGENCE TO MULTI-MODAL DISTRIBUTIONS:
FROM SLOW MIXING TO FAST MIXING
Yongtao Guan and Stephen M. Krone
We compare convergence rates of Metropolis--Hastings chains to multi-
modal
target distributions when the proposal distributions can be of
``local'' and
``small world'' type. In particular, we show that by adding occasional
long-range jumps to a given local proposal distribution, one can turn
a chain
that is ``slowly mixing'' (in the complexity of the problem) into a
chain that
is ``rapidly mixing.'' To do this, we obtain spectral gap estimates
via a new
state decomposition theorem and apply an isoperimetric inequality for
log-concave probability measures. We discuss potential applicability
of our
result to Metropolis-coupled Markov chain Monte Carlo schemes.
http://arXiv.org/abs/math/0703021
---------------------------------------------------------------
5359. TAILS OF RANDOM SUMS OF A HEAVY-TAILED NUMBER OF LIGHT-TAILED
TERMS
Christian Y. Robert and Johan Segers
The tail of the distribution of a sum of a random number of
independent and
identically distributed nonnegative random variables depends on the
tails of
the number of terms and of the terms themselves. This situation is of
interest
in the collective risk model, where the total claim size in a
portfolio is the
sum of a random number of claims. If the tail of the claim number is
heavier
than the tail of the claim sizes, then under certain conditions the
tail of the
total claim size does not change asymptotically if the individual
claim sizes
are replaced by their expectations. The conditions allow the claim
number
distribution to be of consistent variation or to be in the domain of
attraction
of a Gumbel distribution with a mean excess function that grows to
infinity
sufficiently fast. Moreover, the claim number is not necessarily
required to be
independent of the claim sizes.
http://arXiv.org/abs/math/0703022
---------------------------------------------------------------
5360. THE RADIAL SPANNING TREE OF A POISSON POINT PROCESS
Francois Baccelli and Charles Bordenave
We analyze a class of spatial random spanning trees built on a
realization of
a homogeneous Poisson point process of the plane. This tree has a
simple radial
structure with the origin as its root. We first use stochastic geometry
arguments to analyze local functionals of the random tree such as the
distribution of the length of the edges or the mean degree of the
vertices. Far
away from the origin, these local properties are shown to be close to
those of
a variant of the directed spanning tree introduced by Bhatt and Roy.
We then
use the theory of continuous state space Markov chains to analyze
some nonlocal
properties of the tree, such as the shape and structure of its semi-
infinite
paths or the shape of the set of its vertices less than $k$
generations away
from the origin. This class of spanning trees has applications in
many fields
and, in particular, in communications.
http://arXiv.org/abs/math/0703024
---------------------------------------------------------------
5361. RECURRENCE OF EDGE-REINFORCED RANDOM WALK ON A TWO-DIMENSIONAL
GRAPH
Franz Merkl and Silke W.W. Rolles
We consider linearly edge-reinforced random walk on a class of
two-dimensional graphs with constant initial weights. The graphs are
obtained
from Z^2 by replacing every edge by a sufficiently large, but fixed
number of
edges in series. We prove that linearly edge-reinforced random walk
on these
graphs is recurrent. Furthermore, we derive bounds for the
probability that the
edge-reinforced random walk hits the boundary of a large box before
returning
to its starting point.
http://arXiv.org/abs/math/0703027
---------------------------------------------------------------
5362. SELECT SETS: RANK AND FILE
Abba M. Krieger and Moshe Pollak and Ester Samuel-Cahn
In many situations, the decision maker observes items in sequence and
needs
to determine whether or not to retain a particular item immediately
after it is
observed. Any decision rule creates a set of items that are selected. We
consider situations where the available information is the rank of a
present
observation relative to its predecessors. Certain ``natural''
selection rules
are investigated. Theoretical results are presented pertaining to the
evolution
of the number of items selected, measures of their quality and the
time it
would take to amass a group of a given size.
http://arXiv.org/abs/math/0703032
---------------------------------------------------------------
5363. EXISTENCE OF INDEPENDENT RANDOM MATCHING
Darrell Duffie and Yeneng Sun
This paper shows the existence of independent random matching of a large
(continuum) population in both static and dynamic systems, which has
been
popular in the economics and genetics literatures. We construct a joint
agent-probability space, and randomized mutation, partial matching and
match-induced type-changing functions that satisfy appropriate
independence
conditions. The proofs are achieved via nonstandard analysis. The
proof for the
dynamic setting relies on a new Fubini-type theorem for an infinite
product of
Loeb transition probabilities, based on which a continuum of
independent Markov
chains is derived from random mutation, random partial matching and
random type
changing.
http://arXiv.org/abs/math/0703034
---------------------------------------------------------------
5364. EXISTENCE AND UNIQUENESS OF THE MEASURE OF MAXIMAL ENTROPY FOR
THE TEICHMUELLER FLOW ON THE MODULI SPACE OF ABELIAN DIFFERENTIALS
Alexander I. Bufetov and Boris M. Gurevich
We show that the smooth measure is the unique measure of maximal
entropy for
the Teichmueller flow on the moduli space of abelian differentials.
http://arXiv.org/abs/math/0703020
---------------------------------------------------------------
5365. REDUCTIONS AND DEVIATIONS FOR STOCHASTIC PARTIAL DIFFERENTIAL
EQUATIONS UNDER FAST DYNAMICAL BOUNDARY CONDITIONS
Wei Wang and Jinqiao Duan
As a model for multiscale systems under random influences on physical
boundary, a stochastic partial differential equation under a fast random
dynamical boundary condition is investigated. An effective equation
is derived
and justified by reducing the random dynamical boundary condition to
a random
static boundary condition. The effective system is still a stochastic
partial
differential equation, but is more tractable as it is only subject to
the usual
static, instead of dynamical, boundary condition. Furthermore, the
quantitative
comparison between the solution of the original stochastic system and
the
effective solution is provided by proving normal deviations and large
deviations principles. Namely, the normal deviations are shown to be
asymptotically Gaussian, while the rate and speed of the large
deviations are
also determined.
http://arXiv.org/abs/math/0703042
---------------------------------------------------------------
5366. TOLL BASED MEASURES FOR DYNAMICAL GRAPHS
J\'{e}r\'{e}mie Bourdon (LINA) and Damien Eveillard (LINA)
Biological networks are one of the most studied object in computational
biology. Several methods have been developed for studying qualitative
properties of biological networks. Last decade had seen the
improvement of
molecular techniques that make quantitative analyses reachable. One
of the
major biological modelling goals is therefore to deal with the
quantitative
aspect of biological graphs. We propose a probabilistic model that
suits with
this quantitative aspects. Our model combines graph with several
dynamical
sources. It emphazises various asymptotic statistical properties that
might be
useful for giving biological insights
http://arXiv.org/abs/q-bio/0702060
---------------------------------------------------------------
5367. ON THE CHARACTERIZATION OF ISOTROPIC GAUSSIAN FIELDS ON
HOMOGENEOUS SPACES OF COMPACT GROUPS
P.Baldi and D.Marinucci and V.S.Varadarajan
Let T be a random field invariant under the action of a compact group
G We
give conditions ensuring that independence of the random Fourier
coefficients
is equivalent to Gaussianity. As a consequence, in general it is not
possible
to simulate a non-Gaussian invariant random field through its Fourier
expansion
using independent coefficients.
http://arxiv.org/abs/0704.1575
---------------------------------------------------------------
5368. A SYSTEMATIC SCAN FOR 7-COLOURINGS OF THE GRID
Markus Jalsenius and Kasper Pedersen
We study the mixing time of a systematic scan Markov chain for
sampling from
the uniform distribution on proper 7-colourings of a finite rectangular
sub-grid of the infinite square lattice, the grid. A systematic scan
Markov
chain cycles through finite-size subsets of vertices in a
deterministic order
and updates the colours assigned to the vertices of each subset. The
systematic
scan Markov chain that we present cycles through subsets consisting
of 2x2
sub-grids and updates the colours assigned to the vertices using a
procedure
known as heat-bath. We give a computer-assisted proof that this
systematic scan
Markov chain mixes in O(log n) scans, where n is the size of the
rectangular
sub-grid. We make use of a heuristic to compute required couplings of
colourings of 2x2 sub-grids. This is the first time the mixing time of a
systematic scan Markov chain on the grid has been shown to mix for
less than 8
colours. We also give partial results that underline the challenges
of proving
rapid mixing of a systematic scan Markov chain for sampling 6-
colourings of the
grid by considering 2x3 and 3x3 sub-grids.
http://arxiv.org/abs/0704.1625
---------------------------------------------------------------
5369. THE LIL FOR $U$-STATISTICS IN HILBERT SPACES
Rados{\l}aw Adamczak and Rafa{\l} Lata{\l}a
We give necessary and sufficient conditions for the (bounded) law of the
iterated logarithm for $U$-statistics in Hilbert spaces. As a tool we
also
develop moment and tail estimates for canonical Hilbert-space valued
$U$-statistics of arbitrary order, which are of independent interest.
http://arxiv.org/abs/0704.1643
---------------------------------------------------------------
5370. WHERE THE MONOTONE PATTERN (MOSTLY) RULES
Miklos Bona
We consider pattern containment and avoidance with a very tight
definition
that was used first by Riordan more than 60 years ago. Using this
definition,
we prove the monotone pattern is easier to avoid than almost any
other pattern
of the same length.
We also show that with this definition, almost all patterns of
length $k$ are
avoided by the same number of permutations of length $n$. The
corresponding
statements are not known to be true for more relaxed definitions of
pattern
containment. This is the first time we know of that expectations are
used to
compare numbers of permutations avoiding certain patterns.
http://arxiv.org/abs/0704.1489
---------------------------------------------------------------
5371. ASYMPTOTICS OF TRACY-WIDOM DISTRIBUTIONS AND THE TOTAL INTEGRAL
OF A PAINLEV\'E II FUNCTION
Jinho Baik and Robert Buckingham and and Jeffery DiFranco
The Tracy-Widom distribution functions involve integrals of a Painlev
\'e II
function starting from positive infinity. In this paper, we express the
Tracy-Widom distribution functions in terms of integrals starting
from minus
infinity. There are two consequences of these new representations.
The first is
the evaluation of the total integral of the Hastings-McLeod solution
of the
Painlev\'e II equation. The second is the evaluation of the constant
term of
the asymptotic expansions of the Tracy-Widom distribution functions
as the
distribution parameter approaches minus infinity. For the GUE Tracy-
Widom
distribution function, this gives an alternative proof of the recent
work of
Deift, Its, and Krasovsky. The constant terms for the GOE and GSE
Tracy-Widom
distribution functions are new.
http://arxiv.org/abs/0704.3636
---------------------------------------------------------------
5372. INVARIANCE PRINCIPLE FOR ADDITIVE FUNCTIONALS OF MARKOV CHAINS
Yuri N.Kartashov and Alexey M.Kulik
We consider a sequence of additive functionals {\phi_n}, set on a
sequence of
Markov chains {X_n} that weakly converges to a Markov process X. We give
sufficient condition for such a sequence to converge in distribution,
formulated in terms of the characteristics of the additive
functionals, and
related to the Dynkin's theorem on the convergence of W-functionals.
As an
application of the main theorem, the general sufficient condition for
convergence of additive functionals in terms of transition
probabilities of the
chains X_n is proved.
http://arxiv.org/abs/0704.0508
---------------------------------------------------------------
5373. DISSIPATIVE BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS WITH
LOCALLY LIPSCHITZ NONLINEARITY
Fulvia Confortola
In this paper we study a class of backward stochastic differential
equations
(BSDEs) of the form dY(t)= -AY(t)dt -f_0(t,Y(t))dt -f_1(t,Y(t),Z(t))dt +
Z(t)dW(t) on the interval [0,T], with given final condition at time
T, in an
infinite dimensional Hilbert space H. The unbounded operator A is
sectorial and
dissipative and the nonlinearity f_0(t,y) is dissipative and defined
for y only
taking values in a subspace of H. A typical example is provided by the
so-called polynomial nonlinearities. Applications are given to
stochastic
partial differential equations and spin systems.
http://arxiv.org/abs/0704.0509
---------------------------------------------------------------
5374. OPTIMAL CONTROL OF STOCHASTIC DIFFERENTIAL EQUATIONS WITH
DYNAMICAL BOUNDARY CONDITIONS
S. Bonaccorsi and F. Confortola and E. Mastrogiacomo
In this paper we investigate the optimal control problem for a class of
stochastic Cauchy evolution problem with non standard boundary
dynamic and
control. The model is composed by an infinite dimensional dynamical
system
coupled with a finite dimensional dynamics, which describes the boundary
conditions of the internal system. In other terms, we are concerned
with non
standard boundary conditions, as the value at the boundary is
governed by a
different stochastic differential equation.
http://arxiv.org/abs/0704.0524
---------------------------------------------------------------
5375. YIELD CURVE SHAPES AND THE ASYMPTOTIC SHORT RATE DISTRIBUTION
IN AFFINE ONE-FACTOR MODELS
Martin Keller-Ressel and Thomas Steiner
We consider a model for interest rates, where the short rate is given
by a
time-homogenous, one-dimensional affine process in the sense of Duffie,
Filipovic and Schachermayer. We show that in such a model yield
curves can only
be normal, inverse or humped (i.e. endowed with a single local
maximum). Each
case can be characterized by simple conditions on the present short
rate. We
give conditions under which the short rate process will converge to a
limit
distribution and describe the limit distribution in terms of its
cumulant
generating function. We apply our results to the Vasicek model, the
CIR model,
a CIR model with added jumps and a model of Ornstein-Uhlenbeck type.
http://arxiv.org/abs/0704.0567
---------------------------------------------------------------
5376. CONTINUOUS INTERFACES WITH DISORDER: EVEN STRONG PINNING IS TOO
WEAK IN 2 DIMENSIONS
C. Kuelske and E. Orlandi
We consider statistical mechanics models of continuous height effective
interfaces in the presence of a delta-pinning at height zero. There is a
detailed mathematical understanding of the depinning transition in 2
dimensions
without disorder. Then the variance of the interface height w.r.t.
the Gibbs
measure stays bounded uniformly in the volume for any positive
pinning force
and diverges like the logarithm of the pinning force when it tends to
zero.
How does the presence of a quenched disorder term in the
Hamiltonian modify
this transition? We show that an arbitarily weak random field term is
enough to
beat an arbitrarily strong delta-pinning in 2 dimensions and will cause
delocalization. The proof is based on a rigorous lower bound for the
overlap
between local magnetizations and random fields in finite volume. In 2
dimensions it implies growth faster than the volume which is a
contradiction to
localization. We also derive a simple complementary inequality which
shows that
in higher dimensions the fraction of pinned sites converges to one
when the
pinning force tends to infinity.
http://arxiv.org/abs/0704.0582
---------------------------------------------------------------
5377. A NEW APPROACH TO MUTUAL INFORMATION
F. Hiai and D. Petz
A new expression as a certain asymptotic limit via "discrete micro-
states" of
permutations is provided to the mutual information of both continuous
and
discrete random variables.
http://arxiv.org/abs/0704.0588
---------------------------------------------------------------
5378. A NEW APPROACH TO MUTUAL INFORMATION
F. Hiai and D. Petz
A new expression as a certain asymptotic limit via "discrete micro-
states" of
permutations is provided to the mutual information of both continuous
and
discrete random variables.
http://arxiv.org/abs/0704.0588
---------------------------------------------------------------
5379. A PROBABILISTIC REPRESENTATION OF CONSTANTS IN KESTEN'S RENEWAL
THEOREM
Nathana\"{e}l Enriquez (PMA) and Christophe Sabot (ICJ) and Olivier
Zindy (PMA)
The aims of this paper are twofold. Firstly, we derive some
probabilistic
representation for the constant which appears in the one-dimensional
case of
Kesten's renewal theorem. Secondly, we estimate the tail of some
related random
variable which plays an essential role in the description of the
stable limit
law of one-dimensional transient sub-ballistic random walks in random
environment.
http://arXiv.org/abs/math/0703648
---------------------------------------------------------------
5380. LIMIT LAWS FOR TRANSIENT RANDOM WALKS IN RANDOM ENVIRONMENT ON $
\Z$
Nathana\"{e}l Enriquez (PMA) and Christophe Sabot (ICJ) and Olivier
Zindy (PMA)
We consider transient random walks in random environment on $\z$ with
zero
asymptotic speed. A classical result of Kesten, Kozlov and Spitzer
says that
the hitting time of the level $n$ converges in law, after a proper
normalization, towards a positive stable law, but they do not obtain a
description of its parameter. A different proof of this result is
presented,
that leads to a complete characterization of this stable law. The
case of
Dirichlet environment turns out to be remarkably explicit.
http://arXiv.org/abs/math/0703660
---------------------------------------------------------------
5381. COLLISION PROBABILITY FOR RANDOM TRAJECTORIES IN TWO DIMENSIONS
A. Gaudilliere
We give a lower bound for the non-collision probability up to a long
time T
in a system of n independent random walks with fixed obstacles on the
two-dimensional lattice. By `collision' we mean collision between the
random
walks as well as collision with the fixed obstacles. We give an
analogous
result for Brownian particles on the plane. We also explain how this
result can
be used to describe in terms of "quasi random walks" a diluted gas
evolving
under Kawasaki dynamics or simple exclusion.
http://arXiv.org/abs/math/0703671
---------------------------------------------------------------
5382. INFINITE PRODUCTS OF RANDOM MATRICES AND REPEATED INTERACTION
DYNAMICS
Laurent Bruneau and Alain Joye and Marco Merkli
Let $\Psi_n$ be a product of $n$ independent, identically distributed
random
matrices $M$, with the properties that $\Psi_n$ is bounded in $n$,
and that $M$
has a deterministic (constant) invariant vector. Assuming that the
probability
of $M$ having only the simple eigenvalue 1 on the unit circle does
not vanish,
we show that $\Psi_n$ is the sum of a fluctuating and a decaying
process. The
latter converges to zero almost surely, exponentially fast as $n\to
\infty$. The
fluctuating part converges in Cesaro mean to a limit that is
characterized
explicitly by the deterministic invariant vector and the spectral
data of
${\mathbb E}[M]$ associated to 1. No additional assumptions are made
on the
matrices $M$; they may have complex entries and not be invertible.
We apply our general results to two classes of dynamical systems:
inhomogeneous Markov chains with random transition matrices (stochastic
matrices), and random repeated interaction quantum systems. In both
cases, we
prove ergodic theorems for the dynamics, and we obtain the form of
the limit
states.
http://arXiv.org/abs/math/0703675
---------------------------------------------------------------
5383. KOLMOGOROV EQUATIONS FOR MEASURES
Luigi Manca
We consider a semigroup of operators in the Banach space $C_b(H)$ of
uniformly continuous and bounded functions on a separable Hilbert
space $H$. In
particular, we deal with semigroups that are related to solution of
stochastic
PDEs in $H$ and which are not, in general, strongly continuous. We
prove an
existence and uniqueness result for a measure valued equation
involving this
class of semigroups. Then we apply the result to a large class of
second order
differential operators in $C_b(H)$.
http://arXiv.org/abs/math/0703654
---------------------------------------------------------------
5384. APPROXIMATION FOR EXTINCTION PROBABILITY OF THE CONTACT PROCESS
BASED ON THE GR\"OBNER BASIS
Norio Konno
In this note we give a new method for getting a series of
approximations for
the extinction probability of the one-dimensional contact process by
using the
Gr\"obner basis.
http://arXiv.org/abs/0704.0019.abs
---------------------------------------------------------------
5385. CLUSTERING IN A STOCHASTIC MODEL OF ONE-DIMENSIONAL GAS
Vladislav Vysotsky
We give a quantitative analysis of clustering in a stochastic model of
one-dimensional gas. At time zero the gas consists of $n$ identical
particles,
which are randomly distributed on the real line and have zero initial
speeds.
Particles begin to move under the forces of mutual attraction. At a
collision
particles stick together forming a new particle called cluster whose
mass and
speed are defined by the laws of conservation.
We are interested in the asymptotic behaviour of $K_n(t)$ as $n
\to \infty$,
where $K_n(t)$ denotes the number of clusters at time $t$ in the
system with
$n$ initial particles. The main result is a functional limit theorem for
$K_n(t)$. Our proof is based on the discovered localization property
of the
aggregation process. This property states that the behavior of each
particle is
essentially defined only by the motion of neighbour particles.
http://arXiv.org/abs/0704.0086.abs
---------------------------------------------------------------
5386. THE EXACT ASYMPTOTIC OF THE COLLISION TIME TAIL DISTRIBUTION
FOR INDEPENDENT BROWNIAN PARTICLES WITH DIFFERENT DRIFTS
Zbigniew Pucha{\l}a and Tomasz Rolski
In this note we consider the time of the collision $\tau$ for $n$
independent
Brownian motions $X^1_t,...,X_t^n$ with drifts $a_1,...,a_n$, each
starting
from $x=(x_1,...,x_n)$, where $x_1<...<x_n$. We show the exact
asymptotics of
$P_x(\tau>t) = C h(x)t^{-\alpha}e^{-\gamma t}(1 + o(1))$ as $t\to
\infty$ and
identify $C,h(x),\alpha,\gamma$ in terms of the drifts.
http://arXiv.org/abs/0704.0215.abs
---------------------------------------------------------------
5387. PFAFFIANS, HAFNIANS AND PRODUCTS OF REAL LINEAR FUNCTIONALS
P\'eter E. Frenkel
We prove pfaffian and hafnian versions of Lieb's inequalities on
determinants
and permanents of positive semi-definite matrices. We use the hafnian
inequality to improve the lower bound of R\'ev\'esz and Sarantopoulos
on the
norm of a product of linear functionals on a real Euclidean space
(this subject
is sometimes called the `real linear polarization constant' problem).
http://arXiv.org/abs/0704.0028.abs
---------------------------------------------------------------
5388. PINNING AND WETTING TRANSITION FOR (1+1)-DIMENSIONAL FIELDS
WITH LAPLACIAN INTERACTION
Francesco Caravenna and Jean-Dominique Deuschel
We consider a random field \phi: {1, ..., N} -> R as a model for a
linear
chain attracted to the defect line \phi = 0, i.e. the x-axis. The
free law of
the field is specified by the density \exp(-\sum_i V(\Delta \phi_i))
with
respect to the Lebesgue measure on R^N, where \Delta is the discrete
Laplacian
and we allow for a very large class of potentials V(.). The
interaction with
the defect line is introduced by giving the field a reward \epsilon
\ge 0 each
time it touches the x-axis. We call this model the *pinning model*.
We consider
a second model, the *wetting model*, in which, in addition to the
pinning
reward, the field is also constrained to stay non-negative.
We show that both models undergo a phase transition as the
intensity \epsilon
of the pinning reward varies: both in the pinning (a=p) and in the
wetting
(a=w) case, there is a critical value \epsilon_c^a such that when
\epsilon >
\epsilon_c^a the field touches the defect line a positive fraction of
times
(localization), while this does not happen for \epsilon < \epsilon_c^a
(delocalization). The two critical values are non-trivial and
distinct: 0 <
\epsilon_c^p < \epsilon_c^w < \infty, and they are the only non-
analyticity
points of the respective free energies. For the pinning model the
transition is
of second order, hence the field at criticality is delocalized. On
the other
hand, the transition in the wetting model is of first order and the
field at
criticality is localized. The core of our approach is a Markov
renewal theory
description of the field.
http://arXiv.org/abs/math/0703434
---------------------------------------------------------------
5389. TRENDS TO EQUILIBRIUM IN TOTAL VARIATION DISTANCE
Patrick Cattiaux (CMAP and LSProba) and Arnaud Guillin (LATP)
This paper presents different approaches, based on functional
inequalities,
to study the speed of convergence in total variation distance of ergodic
diffusion processes with initial law satisfying a given integrability
condition. To this end, we give a general upper bound "\`{a} la Pinsker"
enabling us to study our problem firstly via usual functional
inequalities
(Poincar\'{e} inequality, weak Poincar\'{e},...) and truncation
procedure, and
secondly through the introduction of new functional inequalities $
\Ipsi$. These
$\Ipsi$-inequalities are characterized through measure-capacity
conditions and
$F$-Sobolev inequalities. A direct study of the decay of Hellinger
distance is
also proposed. Finally we show how a dynamic approach based on
reversing the
role of the semi-group and the invariant measure can lead to interesting
bounds.
http://arXiv.org/abs/math/0703451
---------------------------------------------------------------
5390. CRITICAL BEHAVIOR AND THE LIMIT DISTRIBUTION FOR LONG-RANGE
ORIENTED PERCOLATION. I
Lung-Chi Chen and Akira Sakai
We consider oriented percolation on Z^d times Z_+ whose bond-occupation
probability is pD(...), where p is the percolation parameter and D
(...) is a
probability distribution on Z^d. Suppose that D(x) decays as |x|^{-d-
\alpha}
for some \alpha>0. We prove that the two-point function obeys an
infrared bound
which implies that various critical exponents take on their respective
mean-field values above the upper-critical dimension 2\min{\alpha,2}.
We also
show that the Fourier transform of the normalized two-point function
at time n,
with a proper spatial scaling, has a convergent subsequence to e to
the power
-c|k|^{\min{\alpha,2}} for some c>0.
http://arXiv.org/abs/math/0703455
---------------------------------------------------------------
5391. DOBRUSHIN CONDITIONS FOR SYSTEMATIC SCAN WITH BLOCK DYNAMICS
Kasper Pedersen
We study the mixing time of systematic scan Markov chains on finite spin
systems. It is known that, in a single site setting, the mixing time of
systematic scan can be bounded in terms of the influences sites have
on each
other. We generalise this technique for bounding the mixing time of
systematic
scan to block dynamics, a setting in which a (constant size) set of
sites are
updated simultaneously. In particular we consider the parameter alpha,
corresponding to the maximum influence on any site, and show that if
alpha<1
then the corresponding systematic scan Markov chain mixes rapidly. As
applications of this method we prove O(log n) mixing of systematic
scan (for
any scan order) for heat-bath updates of edges for proper q-
colourings of a
general graph with maximum vertex-degree Delta when q>=2Delta. We
also apply
the method to improve the number of colours required in order to
obtain mixing
in O(log n) scans for systematic scan for heat-bath updates on trees,
using
some suitable block updates.
http://arXiv.org/abs/math/0703461
---------------------------------------------------------------
5392. EFFECTIVE NON-ADDITIVE PAIR POTENTIAL FOR LOCK-AND-KEY INTERACTING
Julio Largo and Piero Tartaglia and Francesco Sciortino
Theoretical studies of self-assembly processes and condensed phases in
colloidal systems are often based on effective inter-particle
potentials. Here
we show that developing an effective potential for particles
interacting with a
limited number of ``lock-and-key'' selective bonds (due to the
specificity of
bio-molecular interactions) requires -- beside the non-sphericity of the
potential -- a (many body) constraint that prevent multiple bonding
on the same
site. We show the importance of retaining both valence and bond-
selectivity by
developing, as a case study, a simple effective potential describing the
interaction between colloidal particles coated by four single-strand DNA
chains.
http://arXiv.org/abs/cond-mat/0703383
---------------------------------------------------------------
5393. DETERMINISTIC RANDOM WALKS ON THE TWO-DIMENSIONAL GRID
Benjamin Doerr and Tobias Friedrich
Jim Propp's rotor router model is a deterministic analogue of a
random walk
on a graph. Instead of distributing chips randomly, each vertex
serves its
neighbors in a fixed order. We analyze the difference between Propp
machine and
random walk on the infinite two-dimensional grid. It is known that,
apart from
a technicality, independent of the starting configuration, at each
time, the
number of chips on each vertex in the Propp model deviates from the
expected
number of chips in the random walk model by at most a constant. We
show that
this constant is approximately 7.8, if all vertices serve their
neighbors in
clockwise or counterclockwise order and 7.3 otherwise. This result in
particular shows that the order in which the neighbors are served
makes a
difference. Our analysis also reveals a number of further unexpected
properties
of the two-dimensional Propp machine.
http://arXiv.org/abs/math/0703453
---------------------------------------------------------------
5394. NON-MONOTONE CONVERGENCE IN THE QUADRATIC WASSERSTEIN DISTANCE
Walter Schachermayer and Uwe Schmock and Josef Teichmann
We give an easy counter-example to Problem 7.20 from C. Villani's
book on
mass transport: in general, the quadratic Wasserstein distance
between $n$-fold
normalized convolutions of two given measures fails to decrease
monotonically.
http://arxiv.org/abs/0704.0876
---------------------------------------------------------------
5395. METROPOLIS ALGORITHM AND EQUIENERGY SAMPLING FOR TWO MEAN FIELD
SPIN SYSTEMS
Bassetti Federico and Leisen Fabrizio
In this paper we study the Metropolis algorithm in connection with two
mean--field spin systems, the so called mean--field Ising model and the
Blume--Emery--Griffiths model. In both this examples the naive choice of
proposal chain gives rise, for some parameters, to a slowly mixing
Metropolis
chain, that is a chain whose spectral gap decreases exponentially
fast (in the
dimension $N$ of the problem). Here we show how a slight variant in the
proposal chain can avoid this problem, keeping the mean computational
cost
similar to the cost of the usual Metropolis. More precisely we prove
that, with
a suitable variant in the proposal, the Metropolis chain has a
spectral gap
which decreases polynomially in 1/N. Using some symmetry structure of
the
energy, the method rests on allowing appropriate jumps within the
energy level
of the starting state.
http://arxiv.org/abs/0704.0906
---------------------------------------------------------------
5396. RANDOM WALKS AND ORTHOGONAL POLYNOMIALS: SOME CHALLENGES
F. Alberto Grunbaum
The study of several naturally arising "nearest neighbours" random walks
benefits from the study of the associated orthogonal polynomials and
their
orthogonality measure. I consider extensions of this approach to a
larger class
of random walks. This raises a number of open problems.
http://arXiv.org/abs/math/0703375
---------------------------------------------------------------
5397. INTERACTING AGENT FEEDBACK FINANCE MODEL
Biao Wu
We consider a financial market model which consists of a financial
asset and
a large number of interacting agents classified into many types.
Different
types of agents are heterogeneous in their price expectations. Each
agent can
change its type based on the current empirical distribution of the
types and
the equilibrium price, and the equilibrium price follows a recursive
price
mechanism based on the previous price and the current empirical
distribution of
the types. The interaction among the agents, and the interaction
between the
agents and the equilibrium price, feedback, are modeled. We analyze the
asymptotic behavior of the empirical distribution of the types and the
equilibrium price when the number of agents goes to infinity. We give
a case
study of a simple example, and also investigate the fixed points of
empirical
distribution and equilibrium price of the example.
http://arXiv.org/abs/math/0703827
---------------------------------------------------------------
5398. A LIMIT THEOREM FOR FINANCIAL MARKETS WITH INERT INVESTORS
Erhan Bayraktar and Ulrich Horst and Ronnie Sircar
We study the effect of investor inertia on stock price fluctuations
with a
market microstructure model comprising many small investors who are
inactive
most of the time. It turns out that semi-Markov processes are tailor
made for
modelling inert investors. With a suitable scaling, we show that when
the price
is driven by the market imbalance, the log price process is
approximated by a
process with long range dependence and non-Gaussian returns
distributions,
driven by a fractional Brownian motion. Consequently, investor
inertia may lead
to arbitrage opportunities for sophisticated market participants. The
mathematical contributions are a functional central limit theorem for
stationary semi-Markov processes, and approximation results for
stochastic
integrals of continuous semimartingales with respect to fractional
Brownian
motion.
http://arXiv.org/abs/math/0703831
---------------------------------------------------------------
5399. QUEUEING THEORETIC APPROACHES TO FINANCIAL PRICE FLUCTUATIONS
Erhan Bayraktar and Ulrich Horst and Ronnie Sircar
One approach to the analysis of stochastic fluctuations in market
prices is
to model characteristics of investor behaviour and the complex
interactions
between market participants, with the aim of extracting consequences
in the
aggregate. This agent-based viewpoint in finance goes back at least
to the work
of Garman (1976) and shares the philosophy of statistical mechanics
in the
physical sciences. We discuss recent developments in market
microstructure
models. They are capable, often through numerical simulations, to
explain many
stylized facts like the emergence of herding behavior, volatility
clustering
and fat tailed returns distributions. They are typically queueing-
type models,
that is, models of order flows, in contrast to classical economic
equilibrium
theories of utility-maximizing, rational, ``representative'' investors.
Mathematically, they are analyzed using tools of functional central
limit
theorems, strong approximations and weak convergence. Our main
examples focus
on investor inertia, a trait that is well-documented, among other
behavioral
qualities, and modelled using semi-Markov switching processes. In
particular,
we show how inertia may lead to the phenomenon of long-range
dependence in
stock prices.
http://arXiv.org/abs/math/0703832
---------------------------------------------------------------
5400. GEOMETRIC BROWNIAN MOTION WITH DELAY: MEAN SQUARE CHARACTERISATION
J. A. D. Appleby and M. Riedle
A geometric Brownian motion with delay is the solution of a stochastic
differential equation where the drift and diffusion coefficient
depend linearly
on the past of the solution, i.e. a linear stochastic functional
differential
equation. In this work the asymptotic behavior in mean square of a
geometric
Brownian motion with delay is completely characterized by a
sufficient and
necessary condition in terms of the drift and diffusion coefficients.
http://arXiv.org/abs/math/0703837
---------------------------------------------------------------
5401. ESTIMATING THE FRACTAL DIMENSION OF THE S&P 500 INDEX USING
WAVELET ANALYSIS
Erhan Bayraktar and H. Vincent Poor and Ronnie Sircar
S&P 500 index data sampled at one-minute intervals over the course of
11.5
years (January 1989- May 2000) is analyzed, and in particular the Hurst
parameter over segments of stationarity (the time period over which
the Hurst
parameter is almost constant) is estimated. An asymptotically
unbiased and
efficient estimator using the log-scale spectrum is employed. The
estimator is
asymptotically Gaussian and the variance of the estimate that is
obtained from
a data segment of $N$ points is of order $\frac{1}{N}$. Wavelet
analysis is
tailor made for the high frequency data set, since it has low
computational
complexity due to the pyramidal algorithm for computing the detail
coefficients. This estimator is robust to additive non-
stationarities, and here
it is shown to exhibit some degree of robustness to multiplicative
non-stationarities, such as seasonalities and volatility persistence,
as well.
This analysis shows that the market became more efficient in the period
1997-2000.
http://arXiv.org/abs/math/0703834
---------------------------------------------------------------
5402. CORRESPONDENCE BETWEEN LIFETIME MINIMUM WEALTH AND UTILITY OF
CONSUMPTION
Erhan Bayraktar and Virginia R. Young
We establish when the two problems of minimizing a function of lifetime
minimum wealth and of maximizing utility of lifetime consumption
result in the
same optimal investment strategy on a given open interval $O$ in
wealth space.
To answer this question, we equate the two investment strategies and
show that
if the individual consumes at the same rate in both problems -- the
consumption
rate is a control in the problem of maximizing utility -- then the
investment
strategies are equal only when the consumption function is linear in
wealth on
$O$, a rather surprising result. It, then, follows that the
corresponding
investment strategy is also linear in wealth and the implied utility
function
exhibits hyperbolic absolute risk aversion.
http://arXiv.org/abs/math/0703820
---------------------------------------------------------------
5403. OPTIMIZING VENTURE CAPITAL INVESTMENTS IN A JUMP DIFFUSION MODEL
Erhan Bayraktar and Masahiko Egami
We study a practical optimization problems for venture capital
investments
and/or Research and Development (R&D) investments. The first problem
is that,
given the amount of the initial investment and the reward function at
the
initial public offering (IPO) market, the venture capitalist wants to
maximize
overall discounted cash flows after subtracting subsequent (if needed)
investments. We describe this problem as a mixture of singular
stochastic
control and optimal stopping problems and give an explicit solution.
The former
corresponds to finding an optimal subsequent investment policy for
the purpose
that the value of the investee company stays away from zero. The latter
corresponds to finding an optimal stopping rule in order to maximize the
harvest of their investments. The second kind problem is concerned about
optimal dividend policy. Rather than selling the holding stock, the
investor
may extract dividends when it is appropriate. We will find a quasi-
explicit
optimal solution to this problem and prove the existence and
uniqueness of the
solution and the optimality of the proposed strategy.
http://arXiv.org/abs/math/0703823
---------------------------------------------------------------
5404. MINIMIZING THE LIFETIME SHORTFALL OR SHORTFALL AT DEATH
Erhan Bayraktar
We find the optimal investment strategy for an individual who seeks to
minimize one of four objectives: (1) the probability that his wealth
reaches a
specified ruin level {\it before} death, (2) the probability that his
wealth
reaches that level {\it at} death, (3) the expectation of how low his
wealth
drops below a specified level {\it before} death, and (4) the
expectation of
how low his wealth drops below a specified level {\it at} death.
Young (2004)
showed that under criterion (1), the optimal investment strategy is a
heavily
leveraged position in the risky asset for low wealth.
In this paper, we introduce the other three criteria in order to
reduce the
leveraging observed by Young (2004). We discovered that surprisingly the
optimal investment strategy for criterion (3) is {\it identical} to
the one for
(1) and that the strategies for (2) and (4) are {\it more} leveraged
than the
one for (1) at low wealth. Because these criteria do not reduce
leveraging, we
completely remove it by considering problems (1) and (3) under the
restriction
that the individual cannot borrow to invest in the risky asset.
http://arXiv.org/abs/math/0703824
---------------------------------------------------------------
5405. OPTIMAL DIVIDEND PAYMENTS UNDER FIXED COST AND IMPLEMENTATION
DELAYS FOR VARIOUS MODELS
Erhan Bayraktar and Masahiko Egami
In this paper we solve the dividend optimization problem for a
corporation or
a financial institution when the managers of the corporation are facing
(regulatory) implementation delays. We consider several cash
reservoir models
for the firm including two mean-reverting processes, Ornstein-
Uhlenbeck and
square-root processes. We provide our solution via a new
characterization of
the value function for one-dimensional diffusions and provide easily
implementable algorithms to find the optimal control and the value
function.
http://arXiv.org/abs/math/0703825
---------------------------------------------------------------
5406. OPTIMAL TIME TO CHANGE PREMIUMS
Erhan Bayraktar and H. Vincent Poor
The claim arrival process to an insurance company is modeled by a
compound
Poisson process whose intensity and/or jump size distribution changes
at an
unobservable time with a known distribution. It is in the insurance
company's
interest to detect the change time as soon as possible in order to re-
evaluate
a new fair value for premiums to keep its profit level the same. This is
equivalent to a problem in which the intensity and the jump size
change at the
same time but the intensity changes to a random variable with a know
distribution. This problem becomes an optimal stopping problem for a
Markovian
sufficient statistic. Here, a special case of this problem is solved,
in which
the rate of the arrivals moves up to one of two possible values, and the
Markovian sufficient statistic is two-dimensional.
http://arXiv.org/abs/math/0703828
---------------------------------------------------------------
5407. THE EFFECTS OF IMPLEMENTATION DELAY ON DECISION-MAKING UNDER
UNCERTAINTY
Erhan Bayraktar and Masahiko Egami
In this paper, we accomplish two objectives: First, we provide a new
mathematical characterization of the value function for impulse control
problems with implementation delay and present a direct solution
method that
differs from its counterparts that use quasi-variational
inequalities. Our
method is direct, in the sense that we do not have to guess the form
of the
solution and we do not have to prove that the conjectured solution
satisfies
conditions of a verification lemma. Second, by employing this direct
solution
method, we solve two examples that involve decision delays: an
exchange rate
intervention problem and a problem of labor force optimization.
http://arXiv.org/abs/math/0703833
---------------------------------------------------------------
5408. MINIMIZING THE PROBABILITY OF LIFETIME RUIN UNDER BORROWING
CONSTRAINTS
Erhan Bayraktar and Virginia R. Young
We determine the optimal investment strategy of an individual who
targets a
given rate of consumption and who seeks to minimize the probability
of going
bankrupt before she dies, also known as {\it lifetime ruin}. We
impose two
types of borrowing constraints: First, we do not allow the individual
to borrow
money to invest in the risky asset nor to sell the risky asset short.
However,
the latter is not a real restriction because in the unconstrained
case, the
individual does not sell the risky asset short. Second, we allow the
individual
to borrow money but only at a rate that is higher than the rate
earned on the
riskless asset.
We consider two forms of the consumption function: (1) The individual
consumes at a constant (real) dollar rate, and (2) the individual
consumes a
constant proportion of her wealth. The first is arguably more
realistic, but
the second is closely connected with Merton's model of optimal
consumption and
investment under power utility. We demonstrate that connection in
this paper,
as well as include a numerical example to illustrate our results.
http://arXiv.org/abs/math/0703850
---------------------------------------------------------------
5409. ON DISCRETE TIME HEDGING IN D-DIMENSIONAL OPTION PRICING MODELS
Mika Hujo
We study the approximation of certain stochastic integrals with
respect to a
d-dimensional diffusion by corresponding stochastic integrals with
piece-wise
constant integrands. In finance this corresponds to replacing a
continuously
adjusted portfolio by discretely adjusted one. The approximation
error is
measured with respect to $L^2$ and it is shown that under certain
assumptions
the approximation rate is $n^{-1/2}$ when one optimizes over
deterministic but
not necessarily equidistant time-nets.
http://arXiv.org/abs/math/0703481
---------------------------------------------------------------
5410. SOLVABILITY OF BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS WITH
QUADRATIC GROWTH
Revaz Tevzadze
We prove the existence of the unique solution of a general Backward
Stochastic Differential Equation with quadratic growth driven by
martingales.
Some kind of comparison theorem is also proved.
http://arXiv.org/abs/math/0703484
---------------------------------------------------------------
5411. ON SOME SPECIAL DIRECTED LAST-PASSAGE PERCOLATION MODELS
Kurt Johansson
We investigate extended processes given by last-passage times in
directed
models defined using exponential variables with decaying mean. In
certain cases
we find the universal Airy process, but other cases lead to non-
universal and
trivial extended processes.
http://arXiv.org/abs/math/0703492
---------------------------------------------------------------
5412. BELIEF PROPAGATION AND BETHE APPROXIMATION FOR TRAFFIC PREDICTION
Cyril Furtlehner (INRIA Futurs) and Jean-Marc Lasgouttes (INRIA
Rocquencourt), Arnaud De La Fortelle (INRIA Rocquencourt)
We define and study an inference algorithm based on "belief
propagation" (BP)
and the Bethe approximation. The idea is to encode into a graph an a
priori
information composed of correlations or marginal probabilities of
variables,
and to use a message passing procedure to estimate the actual state
from some
extra real-time information. This method is originally designed for
traffic
prediction and is particularly suitable in settings where the only
information
available is floating car data. We propose a discretized traffic
description,
based on the Ising model of statistical physics, in order to both
reconstruct
and predict the traffic in real time. General properties of BP are
addressed in
this context. In particular, a detailed study of stability is
proposed with
respect to the a priori data and the graph topology. The behavior of the
algorithm is illustrated by numerical studies on a simple traffic toy
model.
How this approach can be generalized to encode superposition of many
traffic
patterns is discussed.
http://arXiv.org/abs/physics/0703159
---------------------------------------------------------------
5413. RECONSTRUCTION FOR MODELS ON RANDOM GRAPHS
Antoine Gerschenfeld and Andrea Montanari
The reconstruction problem requires to estimate a random variable
given `far
away' observations. Several theoretical results (and simple
algorithms) are
available when the underlying probability distribution is Markov with
respect
to a tree. In this paper we estabilish several exact thresholds for
loopy
graphs. More precisely we consider models on random graphs that converge
locally to trees. We establish the reconstruction thresholds for the
Ising
model both with attractive and random interactions (respectively,
`ferromagnetic' and `spin glass'). Remarkably, in the first case the
result
does not coincide with the corresponding tree threshold.
Among the other tools, we develop a sufficient condition for the
tree and
graph reconstruction problem to coincide. We apply such condition to
antiferromagnetic colorings of random graphs.
http://arxiv.org/abs/0704.3293
---------------------------------------------------------------
5414. ON THE MARGINAL DISTRIBUTIONS OF STATIONARY AR(1) SEQUENCES
S Satheesh and E Sandhya
In this note we correct an omission in our paper (Satheesh and
Sandhya, 2005)
in defining semi-selfdecomposable laws and also show with examples
that the
marginal distributions of a stationary AR(1) process need not even be
infinitely divisible.
http://arxiv.org/abs/0704.3304
---------------------------------------------------------------
5415. A CLASS OF PAIRWISE-INDEPENDENT JOININGS
Elise Janvresse (LMRS) and Thierry De La Rue (LMRS)
We introduce a special class of pairwise-independent self-joinings for a
stationary process: Those for which one coordinate is a continuous
function of
the two others. We investigate which properties on the process the
existence of
such a joining entails. In particular, we prove that if the process is
aperiodic, then it has positive entropy. Our other results suggest
that such
pairwise independent, non-independent self-joinings exist only in
very specific
situations: Essentially when the process is a subshift of finite type
topologically conjugate to a full-shift. This provides an argument in
favor of
the conjecture that 2-fold mixing implies 3-fold-mixing.
http://arxiv.org/abs/0704.3358
---------------------------------------------------------------
5416. ANALYTIC CROSSING PROBABILITIES FOR CERTAIN BARRIERS BY
BROWNIAN MOTION
Nabil Kahale
We calculate crossing probabilities and one-sided last exit time
densities
for a class of moving barriers on an interval [0,T] via Schwartz
distributions.
We derive crossing probabilities and first hitting time densities for
another
class of barriers on [0,T] by proving a Schwartz distribution version
of the
method of images. Analytic expressions for crossing probabilities and
related
densities are given for new explicit and semi-explicit barriers.
http://arxiv.org/abs/0704.2826
---------------------------------------------------------------
5417. GAUSSIAN CONDITIONAL INDEPENDENCE RELATIONS HAVE NO FINITE
COMPLETE CHARACTERIZATION
Seth Sullivant
We show that there can be no finite list of conditional independence
relations which can be used to deduce all conditional independence
implications
among Gaussian random variables. To do this, we construct, for each
$n> 3$ a
family of $n$ conditional independence statements on $n$ random
variables which
together imply that $X_1 \ind X_2$, and such that no subset have this
same
implication. The proof relies on binomial primary decomposition.
http://arxiv.org/abs/0704.2847
---------------------------------------------------------------
5418. CLASSICAL AND QUANTUM RANDOMNESS AND THE FINANCIAL MARKET
Andrei Khrennikov
We analyze complexity of financial (and general economic) processes by
comparing classical and quantum-like models for randomness. Our analysis
implies that it might be that a quantum-like probabilistic
description is more
natural for financial market than the classical one. A part of our
analysis is
devoted to study the possibility of application of the quantum
probabilistic
model to agents of financial market. We show that, although the
direct quantum
(physical) reduction (based on using the scales of quantum mechanics) is
meaningless, one may apply so called quantum-like models. In our
approach
quantum-like probabilistic behaviour is a consequence of contextualy of
statistical data in finances (and economics in general). However, our
hypothesis on "quantumness" of financial data should be tested
experimentally
(as opposed to the conventional description based on the noncontextual
classical probabilistic approach). We present a new statistical test
based on a
generalization of the well known in quantum physics Bell's inequality.
http://arxiv.org/abs/0704.2865
---------------------------------------------------------------
5419. COMPARISON OF SERVICE DISCIPLINES IN REAL-TIME QUEUEING
Pascal Moyal
In this short paper we present a comparison of the service
disciplines in
real-time queueing systems (the customers have a deadline before
which they
should enter the service booth). We state that the more a service
discipline
gives priority to customers having an early deadline, the least the
average
stationary lateness is. We show this result by comparing adequate random
vectors with the Schur-Convex majorization ordering.
http://arxiv.org/abs/0704.2885
---------------------------------------------------------------
5420. THE SPECTRAL LAWS OF HERMITIAN BLOCK-MATRICES WITH LARGE RANDOM
BLOCKS
Tamer Oraby
We are going to study the limiting spectral measure of fixed dimensional
Hermitian block-matrices with large dimensional Wigner blocks. We are
going
also to identify the limiting spectral measure when the Hermitian
block-structure is Circulant. Using the limiting spectral measure of a
Hermitian Circulant block-matrix we will show that the spectral
measure of a
Wigner matrix with $k-$weakly dependent entries need not to be the
semicircle
law in the limit.
http://arxiv.org/abs/0704.2904
---------------------------------------------------------------
5421. LADDER SANDPILES
Antal A. J\'arai and Russell Lyons
We study Abelian sandpiles on graphs of the form $G \times I$, where
$G$ is
an arbitrary finite connected graph, and $I \subset \Z$ is a finite
interval.
We show that for any fixed $G$ with at least two vertices, the
stationary
measures $\mu_I = \mu_{G \times I}$ have two extremal weak limit
points as $I
\uparrow \Z$. The extremal limits are the only ergodic measures of
maximum
entropy on the set of infinite recurrent configurations. We show that
under any
of the limiting measures, one can add finitely many grains in such a
way that
almost surely all sites topple infinitely often. We also show that
the extremal
limiting measures admit a Markovian coding.
http://arxiv.org/abs/0704.2913
---------------------------------------------------------------
5422. UNIQUENESS THRESHOLDS ON TREES VERSUS GRAPHS
Allan Sly
Counter to the general notion that the regular tree is the worst case
for
decay of correlation between sets and nodes we produce an example of a
multi-spin interacting system which has uniqueness on the d-regular
tree but
does not have uniqueness on some infinite d-regular graphs.
http://arxiv.org/abs/0704.2916
---------------------------------------------------------------
5423. HYDRODYNAMIC LIMIT OF EXCLUSION PROCESSES AMONG RANDOM
CONDUCTANCES ON THE SUPERCRITICAL PERCOLATION CLUSTER
A. Faggionato
We prove homogenization results for random walks among random
conductances on
the infinite cluster of bond percolation on Z^d, d>1, with supercritical
parameter p in (p_c, 1]. Conductances are assumed to be bounded
i.i.d. random
variables satisfying an ellipticity condition. As a byproduct,
applying the
general criterium of \cite{F} leading to the hydrodynamic limit of
exclusion
processes with bond-dependent transition rates, we prove for almost all
realizations of the environment the hydrodynamic limit of simple
exclusion
processes among bounded, i.i.d. and elliptic conductances on the
infinite
cluster of supercritical bond percolation. The hydrodynamic equation
is given
by an heat equation whose diffusion coefficient does not depend on the
environment.
http://arxiv.org/abs/0704.3020
---------------------------------------------------------------
5424. THE EVOLUTION OF LARGE COMPONENTS IN RANDOM INDUCED SUBGRAPHS
OF N-CUBES
Christian M. Reidys
In this paper we study random induced subgraphs of binary $n$-cubes,
$Q_2^n$.
This random graph is obtained by selecting each vertex with independent
probability $\lambda_n$. Using a novel construction of sub components
we study
the evolution of the largest component for $\lambda_n=\frac{1+\chi_n}
{n}$,
where $\chi_n$ tends to zero. Our main result is that for $\chi_n=
\epsilon
n^{\frac{a-1}{2}}$, $\epsilon>0$ and arbitrary $1\ge a>0$ there
exists a.s. an
unique largest component of size $\kappa_a n^{a-2} 2^n$, where $
\kappa_a>0$. In
particular in case of $a=1$, i.e. $\lambda_n=\frac{1+\epsilon}{n}$, this
implies the existence of an unique giant component. We can prove our
main
theorem without using Harper's isoperimetric inequality and all
proofs hold
verbatim for generalized $n$-cubes i.e. cubes over an arbitrary finite
alphabet.
http://arxiv.org/abs/0704.2868
---------------------------------------------------------------
5425. STOCHASTIC HEAT EQUATION DRIVEN BY FRACTIONAL NOISE AND LOCAL TIME
Yaozhong Hu and David Nualart
The aim of this paper is to study the $d$-dimensional stochastic heat
equation with a multiplicative Gaussian noise which is white in space
and it
has the covariance of a fractional Brownian motion with Hurst
parameter $% H\in
(0,1)$ in time. Two types of equations are considered. First we
consider the
equation in the It\^{o}-Skorohod sense, and later in the Stratonovich
sense. An
explicit chaos development for the solution is obtained. On the other
hand, the
moments of the solution are expressed in terms of the exponential
moments of
some weighted intersection local time of the Brownian motion.
http://arxiv.org/abs/0704.1824
---------------------------------------------------------------
5426. INFORMATION-BASED ASSET PRICING
Dorje C. Brody and Lane P. Hughston and Andrea Macrina
A new framework for asset price dynamics is introduced in which the
concept
of noisy information about future cash flows is used to derive the price
processes. In this framework an asset is defined by its cash-flow
structure.
Each cash flow is modelled by a random variable that can be expressed
as a
function of a collection of independent random variables called
market factors.
With each such "X-factor" we associate a market information process,
the values
of which are accessible to market agents. Each information process is
a sum of
two terms; one contains true information about the value of the
market factor;
the other represents "noise". The noise term is modelled by an
independent
Brownian bridge. The market filtration is assumed to be that
generated by the
aggregate of the independent information processes. The price of an
asset is
given by the expectation of the discounted cash flows in the risk-
neutral
measure, conditional on the information provided by the market
filtration. When
the cash flows are the dividend payments associated with equities, an
explicit
model is obtained for the share-price, and the prices of options on
dividend-paying assets are derived. Remarkably, the resulting formula
for the
price of a European call option is of the Black-Scholes-Merton type. The
information-based framework also generates a natural explanation for
the origin
of stochastic volatility.
http://arxiv.org/abs/0704.1976
---------------------------------------------------------------
5427. ON A NEW VERSION OF THE ITO'S FORMULA FOR THE STOCHASTIC HEAT
EQUATION
Alberto Lanconelli
We derive an It\^o's-type formula for the one dimensional stochastic
heat
equation driven by a space-time white noise. The proof is based on
elementary
properties of the $\mathcal{S}$-transform and on the explicit
representation of
the solution process. We also discuss the relationship with other
versions of
this It\^o's-type formula existing in literature.
http://arxiv.org/abs/0704.2018
---------------------------------------------------------------
5428. PURE INDUCTIVE LIMIT STATE AND KOLMOGOROV'S PROPERTY
Anilesh Mohari
Let $(\clb,\lambda_t,\psi)$ be a $C^*$-dynamical system where $
(\lambda_t: t
\in \IT_+)$ be a semigroup of injective endomorphism and $\psi$ be an
$(\lambda_t)$ invariant state on the $C^*$ subalgebra $\clb$ and $\IT_
+$ is
either non-negative integers or real numbers. The central aim of this
exposition is to find a useful criteria for the inductive limit state
$\clb
\raro^{\lambda_t} \clb$ canonically associated with $\psi$ to be
pure. We
achieve this by exploring the minimal weak forward and backward Markov
processes associated with the Markov semigroup on the corner von-Neumann
algebra of the support projection of the state $\psi$ to prove that
Kolmogorov's property [Mo2] of the Markov semigroup is a sufficient
condition
for the inductive state to be pure. As an application of this
criteria we find
a sufficient condition for a translation invariant factor state on a one
dimensional quantum spin chain to be pure. This criteria in a sense
complements
criteria obtained in [BJKW,Mo2] as we could go beyond lattice
symmetric states.
http://arxiv.org/abs/0704.1987
---------------------------------------------------------------
5429. JONES INDEX OF A QUANTUM DYNAMICAL SEMIGROUP
Anilesh Mohari
In this paper we consider a semigroup of completely positive maps
$\tau=(\tau_t,t \ge 0)$ with a faithful normal invariant state $\phi$
on a
type-$II_1$ factor $\cla_0$ and propose an index theory. We :achieve
this via a
more general Kolmogorov's type of construction for stationary Markov
processes
which naturally associate a nested isomorphic von-Neumann algebras. In
particular this construction generalizes well known Jones construction
associated with a sub-factor of type-II$_1$ factor.
http://arxiv.org/abs/0704.1989
---------------------------------------------------------------
5430. FRUSTRATION SOLITAIRE
Peter G. Doyle and Charles M. Grinstead and J. Laurie Snell
In this expository article, we discuss the rank-derangement problem,
which
asks for the number of permutations of a deck of cards such that each
card is
replaced by a card of a different rank. This combinatorial problem
arises in
computing the probability of winning the game of `frustration
solitaire'. We
discuss and exhibit the solution to a related problem, Montmort's
`Probleme du
Treize', which dates back to circa 1708.
http://arXiv.org/abs/math/0703900
---------------------------------------------------------------
5431. CONNECTIVITY AND EQUILIBRIUM IN RANDOM GAMES
Constantinos Daskalakis and Alexandros G. Dimakis and Elchanan Mossel
We study how the structure of the interaction graph affects the Nash
equilibria of the resulting game. In particular, for a fixed
interaction graph,
we are interested if there exist Nash equilibria which arise when random
utility tables are assigned to the players.
We provide conditions for the structure of the graph under which
equilibria
are likely to exist and complementary conditions which make the
existence of
equilibria highly unlikely. Our results have immediate implications
for many
deterministic graphs and generalize known results for games on the
complete
graph. In particular, our results imply that the probability that
bounded
degree graphs have Nash equilibria is exponentially small in the size
of the
graph and yield a simple algorithm that finds small non-existence
certificates
for a large family of graphs.
In order to obtained a refined characterization of the degree of
connectivity
associated with the existence of equilibria, we study the model in
the random
graph setting. In particular, we look at the case where the
interaction graph
is drawn from the Erd\H{o}s-R\'enyi, $G(n,p)$, where each edge is
present
independently with probability $p$. For this model we establish a
{\em double
phase transition} for the existence of pure Nash equilibria as a
function of
the average degree $p n$ consistent with the non-monotone behavior of
the
model. We show that when the average degree satisfies $n p > (2 +
\Omega(1))
\log n$, the number of pure Nash equilibria follows a Poisson
distribution with
parameter 1. When $1/n << n p < (0.5 -\Omega(1)) \log n$ pure Nash
equilibria
fail to exist with high probability. Finally, when $n p << 1/n$ a
pure Nash
equilibrium exists with high probability.
http://arXiv.org/abs/math/0703902
---------------------------------------------------------------
5432. ON LERCH'S TRANSCENDENT AND THE GAUSSIAN RANDOM WALK
A. J. E. M. Janssen and J. S. H. van Leeuwaarden
Let $X_1,X_2,...$ be independent variables, each having a normal
distribution
with negative mean $-\beta<0$ and variance 1. We consider the partial
sums
$S_n=X_1+...+X_n$, with $S_0=0$, and refer to the process $\{S_n:n
\geq0\}$ as
the Gaussian random walk. We present explicit expressions for the
mean and
variance of the maximum $M=\max\{S_n:n\geq0\}.$ These expressions are
in terms
of Taylor series about $\beta=0$ with coefficients that involve the
Riemann
zeta function. Our results extend Kingman's first-order approximation
[Proc.
Symp. on Congestion Theory (1965) 137--169] of the mean for $\beta
\downarrow0$.
We build upon the work of Chang and Peres [Ann. Probab. 25 (1997)
787--802],
and use Bateman's formulas on Lerch's transcendent and Euler--Maclaurin
summation as key ingredients.
http://arXiv.org/abs/math/0703908
---------------------------------------------------------------
5433. EFFICIENT IMPORTANCE SAMPLING FOR MONTE CARLO EVALUATION OF
EXCEEDANCE PROBABILITIES
Hock Peng Chan and Tze Leung Lai
Large deviation theory has provided important clues for the choice of
importance sampling measures for Monte Carlo evaluation of exceedance
probabilities. However, Glasserman and Wang [Ann. Appl. Probab. 7 (1997)
731--746] have given examples in which importance sampling measures
that are
consistent with large deviations can perform much worse than direct
Monte
Carlo. We address this problem by using certain mixtures of
exponentially
twisted measures for importance sampling. Their asymptotic optimality is
established by using a new class of likelihood ratio martingales and
renewal
theory.
http://arXiv.org/abs/math/0703910
---------------------------------------------------------------
5434. LOCALIZATION TRANSITION IN DISORDERED PINNING MODELS. EFFECT
OF RANDOMNESS ON THE CRITICAL PROPERTIES
F. Toninelli (Laboratoire de Physique and ENS Lyon and CNRS UMR 5672)
These notes are devoted to the statistical mechanics of directed
polymers
interacting with one-dimensional spatial defects. We are interested in
particular in the situation where frozen disorder is present. These
polymer
models undergo a localization/delocalization transition. There is a
large
(bio)-physics literature on the subject since these systems describe,
for
instance, the statistics of thermally created loops in DNA double
strands and
the interaction between (1+1)-dimensional interfaces and disordered
walls. In
these cases the transition corresponds, respectively, to the DNA
denaturation
transition and to the wetting transition. More abstractly, one may
see these
models as random and inhomogeneous perturbations of renewal processes.
The last few years have witnessed a great progress in the
mathematical
understanding of the equilibrium properties of these systems. In
particular,
many rigorous results about the location of the critical point, about
critical
exponents and path properties of the polymer in the two thermodynamic
phases
(localized and delocalized) are now available.
Here, we will focus on some aspects of this topic - in particular,
on the
non-perturbative effects of disorder. The mathematical tools employed
range
from renewal theory to large deviations and, interestingly, show tight
connections with techniques developed recently in the mathematical
study of
mean field spin glasses.
http://arXiv.org/abs/math/0703912
---------------------------------------------------------------
5435. RANDOMLY GROWING BRAID ON THREE STRANDS AND THE MANTA RAY
Jean Mairesse and Fr\'{e}d\'{e}ric Math\'{e}us
Consider the braid group $B_3=< a,b| aba=bab>$ and the nearest neighbor
random walk defined by a probability $\nu$ with support
$\{a,a^{-1},b,b^{-1}\}$. The rate of escape of the walk is explicitly
expressed
in function of the unique solution of a set of eight polynomial
equations of
degree three over eight indeterminates. We also explicitly describe the
harmonic measure of the induced random walk on $B_3$ quotiented by
its center.
The method and results apply, mutatis mutandis, to nearest neighbor
random
walks on dihedral Artin groups.
http://arXiv.org/abs/math/0703913
---------------------------------------------------------------
5436. BETTI NUMBERS OF RANDOM MANIFOLDS
Michael Farber and Thomas Kappeler
We study mathematical expectations of Betti numbers of configuration
spaces
of planar linkages, viewing the lengths of the bars of the linkage as
random
variables. Our main result gives an explicit asymptotic formulae for
these
mathematical expectations for two distinct probability measures
describing the
statistics of the length vectors when the number of links tends to
infinity. In
the proof we use a combination of geometric and analytic tools. The
average
Betti numbers are expressed in terms of volumes of intersections of a
simplex
with certain half-spaces.
http://arXiv.org/abs/math/0703929
---------------------------------------------------------------
5437. CHUNG'S LAW FOR HOMOGENEOUS BROWNIAN FUNCTIONALS
Aim\'e Lachal (ICJ) and Thomas Simon (DP)
Consider the first exit time $T_{a,b}$ from a finite interval $[-a,b]
$ for an
homogeneous fluctuating functional $X$ of a linear Brownian motion.
We show the
existence of a finite positive constant $\k$ such that
$$\lim_{t\to\infty}t^{-1}\log \p[ T_{ab} > t] = -\k.$$ Following Chung's
original approach, we deduce a "liminf" law of the iterated logarithm
for the
two-sided supremum of $X$. This extends and gives a new point of view
on a
result of Khoshnevisan and Shi.
http://arxiv.org/abs/0704.3519
---------------------------------------------------------------
5438. RAPID MIXING OF GIBBS SAMPLING ON GRAPHS THAT ARE SPARSE ON
AVERAGE
Elchanan Mossel and Allan Sly
In this work we show that for every $d < \infty$ and the Ising model
defined
on $G(n,d/n)$, there exists a $\beta_d > 0$, such that for all $\beta <
\beta_d$ with probability going to 1 as $n \to \infty$, the mixing
time of the
dynamics on $G(n,d/n)$ is polynomial in $n$. Our results are the first
polynomial time mixing results proven for a natural model on $G(n,d/n)
$ for $d
> 1$ where the parameters of the model do not depend on $n$. They
also provide
a rare example where one can prove a polynomial time mixing of Gibbs
sampler in
a situation where the actual mixing time is slower than $n \polylog(n)
$. Our
proof exploits in novel ways the local treelike structure of Erd\H{o}
s-R\'enyi
random graphs, comparison and block dynamics arguments and a recent
result of
Weitz.
Our results extend to much more general families of graphs which
are sparse
in some average sense and to much more general interactions. In
particular,
they apply to any graph for which every vertex $v$ of the graph has a
neighborhood $N(v)$ of radius $O(\log n)$ in which the induced sub-
graph is a
tree union at most $O(\log n)$ edges and where for each simple path
in $N(v)$
the sum of the vertex degrees along the path is $O(\log n)$.
Moreover, our
result apply also in the case of arbitrary external fields and
provide the
first FPRAS for sampling the Ising distribution in this case. We finally
present a non Markov Chain algorithm for sampling the distribution
which is
effective for a wider range of parameters. In particular, for $G(n,d/
n)$ it
applies for all external fields and $\beta < \beta_d$, where $d \tanh
(\beta_d)
= 1$ is the critical point for decay of correlation for the Ising
model on
$G(n,d/n)$.
http://arxiv.org/abs/0704.3603
---------------------------------------------------------------
5439. THE UPPER ENVELOPE OF POSITIVE SELF-SIMILAR MARKOV PROCESSES
Juan Carlos Pardo Millan
We establish integral tests and laws of the iterated logarithm at 0
and at
$+\infty$, for the upper envelope of positive self-similar Markov
processes.
Our arguments are based on the Lamperti representation, time reversal
arguments
and on the study of the upper envelope of their future infimum due to
Pardo
\cite{Pa}. These results extend integral test and laws of the iterated
logarithm for Bessel processes due to Dvoretsky and Erd\"os \cite{de}
and
stable L\'evy processes conditioned to stay positive with no positive
jumps due
to Bertoin \cite{be1}.
http://arXiv.org/abs/math/0703071
---------------------------------------------------------------
5440. EXISTENCE AND SPATIAL LIMIT THEOREMS FOR LATTICE AND CONTINUUM
PARTICLE SYSTEMS
Mathew D. Penrose
We give a general existence result for interacting particle systems with
local interactions and bounded jump rates but noncompact state space
at each
site. We allow for jump events at a site that affect the state of its
neighbours. We give a law of large numbers and functional central
limit theorem
for additive set functions taken over an increasing family of
subcubes of
$Z^d$. We discuss application to marked spatial point processes with
births,
deaths and jumps of particles, in particular examples such as
continuum and
lattice ballistic deposition and a sequential model for random loose
sphere
packing.
http://arXiv.org/abs/math/0703072
---------------------------------------------------------------
5441. BID-ASK DYNAMIC PRICING IN FINANCIAL MARKETS WITH TRANSACTION
COSTS AND LIQUIDITY RISK
Jocelyne Bion-Nadal
We introduce, in continuous time, an axiomatic approach to assign to any
financial position a dynamic ask (resp. bid) price process. Taking
into account
both transaction costs and liquidity risk this leads to the convexity
(resp.
concavity) of the ask (resp. bid) price. Time consistency is a
crucial property
for dynamic pricing. Generalizing the result of Jouini and Kallal, we
prove
that the No Free Lunch condition for a time consistent dynamic pricing
procedure (TCPP) is equivalent to the existence of an equivalent
probability
measure $R$ that transforms a process between the bid process and the
ask
process of any financial instrument into a martingale. Furthermore we
prove
that the ask price process associated with any financial instrument
is then a
$R$-supermartingale process which has a cadlag modification. Finally
we show
that time consistent dynamic pricing allows both to extend the
dynamics of some
reference assets and to be consistent with any observed bid ask
spreads that
one wants to take into account. It then provides new bounds reducing
the bid
ask spreads for the other financial instruments.
http://arXiv.org/abs/math/0703074
---------------------------------------------------------------
5442. DONSKER THEOREM FOR THE ROSENBLATT PROCESS AND A BINARY MARKET
MODEL
Ciprian Tudor (CES and SAMOS) and Soledad Torres
In this paper, we prove a Donsker type approximation theorem for the
Rosenblatt process, which is a selfsimilar stochastic process
exhibiting long
range dependence. By using numerical results and simulated data, we
show that
this approximation performs very well. We use this result to
construct a binary
market model driven by this process and we show that the model admits
arbitrage
opportunities.
http://arXiv.org/abs/math/0703085
---------------------------------------------------------------
5443. MULTIDIMENSIONAL BIFRACTIONAL BROWNIAN MOTION: ITO AND TANAKA
FORMULAS
Ciprian Tudor (CES and SAMOS) and Khalifa Es-Sebaiy
Using the Malliavin calculus with respect to Gaussian processes and the
multiple stochastic integrals we derive It\^{o}'s and Tanaka's
formulas for the
$d$-dimensional bifractional Brownian motion.
http://arXiv.org/abs/math/0703087
---------------------------------------------------------------
5444. THE STOCHASTIC HEAT EQUATION WITH A FRACTIONAL-COLORED NOISE:
EXISTENCE OF THE SOLUTION
Raluca Balan and Ciprian Tudor (CES and SAMOS)
In this article we consider the stochastic heat equation $u_{t}-
\Delta u=\dot
B$ in $(0,T) \times \bR^d$, with vanishing initial conditions, driven
by a
Gaussian noise $\dot B$ which is fractional in time, with Hurst index
$H \in
(1/2,1)$, and colored in space, with spatial covariance given by a
function
$f$. Our main result gives the necessary and sufficient condition on
$H$ for
the existence of the process solution. When $f$ is the Riesz kernel
of order
$\alpha \in (0,d)$ this condition is $H>(d-\alpha)/4$, which is a
relaxation of
the condition $H>d/4$ encountered when the noise $\dot B$ is white in
space.
When $f$ is the Bessel kernel or the heat kernel, the condition remains
$H>d/4$.
http://arXiv.org/abs/math/0703088
---------------------------------------------------------------
5445. ON THE REGULARITY OF STOCHASTIC CURRENTS, FRACTIONAL BROWNIAN
MOTION AND APPLICATIONS TO A TURBULENCE MODEL
Franco Flandoli (DIPARTIMENTO Di Matematica Applicata Pisa) and
Massimiliano Gubinelli (LM-Orsay), Francesco Russo (LAGA)
We study the pathwise regularity of the map $$ \phi \mapsto I(\phi) =
\int_0^T < \phi(X_t), dX_t>$$ where $\phi$ is a vector function on $
\R^d$
belonging to some Banach space $V$, $X$ is a stochastic process and the
integral is some version of a stochastic integral defined via
regularization. A
\emph{stochastic current} is a continuous version of this map, seen
as a random
element of the topological dual of $V$. We give sufficient conditions
for the
current to live in some Sobolev space of distributions and we provide
elements
to conjecture that those are also necessary. Next we verify the
sufficient
conditions when the process $X$ is a $d$-dimensional fractional
Brownian motion
(fBm); we identify regularity in Sobolev spaces for fBm with Hurst
index $H \in
(1/4,1)$. Next we provide some results about general Sobolev
regularity of
Brownian currents. Finally we discuss applications to a model of
random vortex
filaments in turbulent fluids.
http://arXiv.org/abs/math/0703100
---------------------------------------------------------------
5446. ON THE SPECTRAL NORM OF A RANDOM TOEPLITZ MATRIX
Mark W. Meckes
Suppose that $T_n$ is a Toeplitz matrix whose entries come from a
sequence of
independent but not necessarily identically distributed random
variables with
mean zero. Under some additional moment conditions, we show that the
spectral
norm of $T_n$ is of the order $\sqrt{n \log n}$. The same result
holds for
random Hankel matrices as well as other variants of random Toeplitz
matrices
which have been studied in the literature.
http://arXiv.org/abs/math/0703134
---------------------------------------------------------------
5447. EQUILIBRIUM STATES OF TWO STOCHASTIC MODELS IN MATHEMATICAL
ECOLOGY
Feng Yu
This work deals with two problems arising in mathematical ecology.
The first
problem is concerned with diploid branching particle models and its
behavior
when rapid stirring is added to the interaction. The particle models
involve
two types of particles, male and female, and branching can only occur
when both
types of particles are present. We show that if the branching rate is
sufficiently large, this particle model has a nontrivial stationary
distribution, i.e. one that does not concentrate all weight on the
all-0 state,
using a comparison argument due to R. Durrett. We also show
extinction for
small branching rates, thereby establishing the existence of a phase
transition. We then add two different rapid stirring mechanisms to the
interactions and show that for the particle models with rapid
stirring, there
also exist nontrivial stationary distribution(s); for this, we
analyze the
limiting PDE and establish a condition on the PDE that guarantees
existence of
nontrivial stationary distributions for sufficient fast stirring.
The second problem deals with a model of sympatric speciation, i.e.
speciation in the absence of geographical separation, originally
proposed by U.
Dieckmann and M. Doebeli in 1999. We modify their original model to
obtain
several constant-population particle models. We concentrate on a
continuous-time model that converges to a deterministic dynamical
system as the
number of particles becomes large. We establish various results
regarding
whether speciation occurs by studying the existence of bimodal
stationary
distributions for the limiting dynamical system.
http://arXiv.org/abs/math/0703135
---------------------------------------------------------------
5448. MODERATE DEVIATIONS FOR LOG-LIKE FUNCTIONS OF STATIONARY
GAUSSIAN PROCESSES
Boris Tsirelson
A moderate deviation principle for nonlinear functions of Gaussian
processes
is established. The nonlinear functions need not be locally bounded.
Especially, the logarithm is allowed. (Thus, small deviations of the
process
are relevant.) Both discrete and continuous time is treated. An
integrable
power-like decay of the correlation function is assumed.
http://arXiv.org/abs/math/0703289
---------------------------------------------------------------
5449. SEPARATION CUTOFFS FOR RANDOM WALK ON IRREDUCIBLE REPRESENTATIONS
Jason Fulman
Random walk on the irreducible representations of the symmetric and
general
linear groups is studied. A separation distance cutoff is proved and
the exact
separation distance asymptotics are determined. A key tool is a
method for
writing the multiplicities in the Kronecker tensor powers of a fixed
representation as a sum of non-negative terms. Connections are made
with the
Lagrange-Sylvester interpolation approach to Markov chains.
http://arXiv.org/abs/math/0703291
---------------------------------------------------------------
5450. THE CONDITION NUMBER OF A RANDOMLY PERTURBED MATRIX
Terence Tao and Van Vu
Let $M$ be an arbitrary $n$ by $n$ matrix. We study the condition
number a
random perturbation $M+N_n$ of $M$, where $N_n$ is a random matrix.
It is shown
that, under very general conditions on $M$ and $M_n$, the condition
number of
$M+N_n$ is polynomial in $n$ with very high probability. The main
novelty here
is that we allow $N_n$ to have discrete distribution.
http://arXiv.org/abs/math/0703307
---------------------------------------------------------------
5451. A PROOF OF A NON-COMMUTATIVE CENTRAL LIMIT THEOREM BY THE
LINDEBERG METHOD
Vladislav Kargin
A Central Limit Theorem for non-commutative random variables is
proved using
the Lindeberg method. The theorem is a generalization of the Central
Limit
Theorem for free random variables proved by Voiculescu. The Central
Limit
Theorem in this paper relies on an assumption which is weaker than
freeness.
http://arXiv.org/abs/math/0703345
---------------------------------------------------------------
5452. NON-NEGATIVE INTEGER-VALUED SEMI-SELFSIMILAR PROCESSES
S Satheesh and E Sandhya
Non-negative integer-valued semi-selfsimilar processes are
introduced. Levy
processes in this class are characterized. Its relation to an AR(1)
scheme is
derived.
http://arXiv.org/abs/math/0703346
---------------------------------------------------------------
5453. RATE OF CONVERRGENCE FOR ERGODIC CONTINUOUS MARKOV PROCESSES :
LYAPUNOV VERSUS POINCARE
Dominique Bakry (LSProba) and Patrick Cattiaux (MODAL'X and CMAP)
and Arnaud Guillin (LATP)
We study the relationship between two classical approaches for
quantitative
ergodic properties : the first one based on Lyapunov type controls and
popularized by Meyn and Tweedie, the second one based on functional
inequalities (of Poincar\'e type). We show that they can be linked
through new
inequalities (Lyapunov-Poincar\'e inequalities). Explicit examples for
diffusion processes are studied, improving some results in the
literature. The
example of the kinetic Fokker-Planck equation recently studied by H
\'erau-Nier,
Helffer-Nier and Villani is in particular discussed in the final
section.
http://arXiv.org/abs/math/0703355
---------------------------------------------------------------
5454. IS THE UNIVERSE NOISE-SENSITIVE?
Gil Kalai
The dichotomy between noise-stable and (completely) noise-sensitive
stochastic models is of recent interest in probability theory. Of
particular
interest is the study of lattice models coming from statistical
physics. The
Fourier transform of noise-sensitive lattice models is concentrated
on high
eigenvalues and is described by "large" stochastic geometric objects.
Noise
sensitivity occurs quite surprisingly in various models like critical
percolation, and is forced by certain symmetry conditions.
It appears that basic models from high-energy physics are noise
stable; This
is the impression from the basic mathematical frameworks used for
describing
them, and also from the description in terms of particles and
interactions
involving a small number of particles.
More general stochastic models with noise-sensitive components
will not make
a difference in measurements involving particles and their
interactions, but
may provide additional modeling power to proceed where current models
are
insufficient.
http://arXiv.org/abs/hep-th/0703092
---------------------------------------------------------------
5455. DOES THERE EXIST THE LEBESGUE MEASURE IN THE INFINITE-
DIMENSIONAL SPACE?
Anatly Vershik
We consider the sigma-finite measures in the space of vector-valued
distributions on the manifold $X$ with Laplace transform
$$\Psi(f)=\exp\{-\theta\int_X\ln||f(x)||dx\}, \theta>0.$$
We prove that the weak limit of Haar measures on the Cartan
subgroup of the
group $SL(n,{\Bbb R})$ when $n$ tends to infinity is just that
measure which we
called infinite dimensional Lebesgue measure.
This measure is invariant under the linear action of some
infinite-dimensional Abelian group. Application to the representation
theory of
the current groups was one of the reason to define this measure. The
measure
also is closely related to the Poisson--Dirichlet measures well known in
combinatorics and probability theory. The only known example of the
analogous
asymptotical behavior of the uniform measure on the homogeneous
manifold is
{\it classical Maxwell-Poincar\'e lemma} which asserts that the weak
limit of
uniform measures on the Euclidean sphere of appropriate radius as
dimension
tends to infinity is the standard infinite-dimensional Gaussian
measure. In our
situation all the measures are no more finite but sigma-finite.
http://arXiv.org/abs/math-ph/0703033
---------------------------------------------------------------
5456. APPROXIMATION OF QUANTUM LEVY PROCESSES BY QUANTUM RANDOM WALKS
Uwe Franz and Adam Skalski
Every quantum Levy process with a bounded stochastic generator is
shown to
arise as a strong limit of a family of suitably scaled quantum random
walks.
http://arXiv.org/abs/math/0703339
---------------------------------------------------------------
5457. ON A REMARKABLE SEMIGROUP OF HOMOMORPHISMS WITH RESPECT TO
FREE MULTIPLICATIVE CONVOLUTION
Serban T. Belinschi and Alexandru Nica
Let M denote the space of Borel probability measures on the real
line. For
every nonnegative t we consider the transformation $\mathbb B_t : M
\to M$
defined for any given element in M by taking succesively the the (1
+t) power
with respect to free additive convolution and then the 1/(1+t) power
with
respect Boolean convolution of the given element. We show that the
family of
maps {\mathbb B_t|t\geq 0} is a semigroup with respect to the
operation of
composition and that, quite surprisingly, every $\mathbb B_t$ is a
homomorphism
for the operation of free multiplicative convolution.
We prove that for t=1 the transformation $\mathbb B_1$ coincides
with the
canonical bijection $\mathbb B : M \to M_{inf-div}$ discovered by
Bercovici and
Pata in their study of the relations between infinite divisibility in
free and
in Boolean probability. Here M_{inf-div} stands for the set of
probability
distributions in M which are infinitely divisible with respect to
free additive
convolution. As a consequence, we have that $\mathbb B_t(\mu)$ is
infinitely
divisible with respect to free additive convolution for any for every
$\mu$ in
M and every t greater than or equal to one.
On the other hand we put into evidence a relation between the
transformations
$\mathbb B_t$ and the free Brownian motion; indeed, Theorem 4 of the
paper
gives an interpretation of the transformations $\mathbb B_t$ as a way of
re-casting the free Brownian motion, where the resulting process becomes
multiplicative with respect to free multiplicative convolution, and
always
reaches infinite divisibility with respect to free additive
convolution by the
time t=1.
http://arXiv.org/abs/math/0703295
---------------------------------------------------------------
5458. LARGE DEVIATIONS FOR PARTITION FUNCTIONS OF DIRECTED POLYMERS
AND SOME OTHER MODELS IN AN IID FIELD
Iddo Ben-Ari
Consider the partition function of a directed polymer in an IID
field. We
assume that both tails of the negative and the positive part of the
field are
at least as light as exponential. It is a well-known fact that the
free energy
of the polymer is equal to a deterministic constant for almost every
realization of the field and that the upper tail of the large
deviations is
exponential. The lower tail of the large deviations is typically
lighter than
exponential. In this paper we provide a method to obtain estimates on
the rate
of decay of the lower tail of the large deviations, which are sharp
up to
multiplicative constants. As a consequence, we show that the lower
tail of the
large deviations exhibits three regimes, determined according to the
tail of
the negative part of the field. Our method is simple to apply and can
be used
to cover other oriented and non-oriented models including first/last-
passage
percolation and the parabolic Anderson model
http://arxiv.org/abs/0704.3758
---------------------------------------------------------------
5459. ON THE NUMBER OF COLLISIONS IN $\LAMBDA$-COALESCENTS
Alexander Gnedin and Yuri Yakubovich
We examine the total number of collisions $C_n$ in the $\Lambda$-
coalescent
process which starts with $n$ particles. A linear growth and a stable
limit law
for $C_n$ are shown under the assumption of a power-like behaviour of
the
measure $\Lambda$ near 0 with exponent $0<\alpha<1$.
http://arxiv.org/abs/0704.3902
---------------------------------------------------------------
5460. SMOOTHNESS OF THE LAW OF SOME ONE-DIMENSIONAL JUMPING S.D.E.S
WITH NON-CONSTANT RATE OF JUMP
Nicolas Fournier
We consider a one-dimensional jumping Markov process $\{X^x_t\}_{t
\geq 0}$,
solving a Poisson-driven stochastic differential equation. We prove
that the
law of $X^x_t$ admits a smooth density for $t>0$, under some
regularity and
non-degeneracy assumptions on the coefficients of the S.D.E. To our
knowledge,
our result is the first one including the important case of a non-
constant rate
of jump. The main difficulty is that in such a case, the map $x
\mapsto X^x_t$
is not smooth. This seems to make impossible the use of Malliavin
calculus
techniques. To overcome this problem, we introduce a new method, in
which the
propagation of the smoothness of the density is obtained by analytic
arguments.
http://arxiv.org/abs/0704.3922
---------------------------------------------------------------
5461. DYNAMIC PROGRAMMING PRINCIPLE FOR ONE KIND OF STOCHASTIC
RECURSIVE OPTIMAL CONTROL PROBLEM AND HAMILTON-JACOBI-BELLMAN EQUATIONS
Zhen Wu and Zhiyong Yu
In this paper, we study one kind of stochastic recursive optimal control
problem with the obstacle constraints for the cost function where the
cost
function is described by the solution of one reflected backward
stochastic
differential equations. We will give the dynamic programming
principle for this
kind of optimal control problem and show that the value function is
the unique
viscosity solution of the obstacle problem for the corresponding
Hamilton-Jacobi-Bellman equations.
http://arxiv.org/abs/0704.3775
---------------------------------------------------------------
5462. THE SCALING LIMIT OF FOMIN'S IDENTITY FOR TWO PATHS
Michael J. Kozdron (University of Regina)
We review some recently completed research that establishes the
scaling limit
of Fomin's identity for loop-erased random walk on Z^2, and in the
case of two
paths prove directly that the corresponding identity holds for
chordal SLE(2).
http://arXiv.org/abs/math/0703615
---------------------------------------------------------------
5463. QUOTIENT PROBABILISTIC NORMED SPACES AND COMPLETENESS RESULTS
Bernardo Lafuerza-Guillen and Donal O'Regan and Reza Saadati
We introduce the concept of quotient in PN spaces and give some
examples. We
prove some theorems with regard to the completeness of a quotient.
http://arXiv.org/abs/math/0703629
---------------------------------------------------------------
5464. DYNAMICS OF POSTCRITICALLY BOUNDED POLYNOMIAL SEMIGROUPS
Hiroki Sumi
We investigate the dynamics of semigroups generated by a family of
polynomial
maps on the Riemann sphere such that the postcritical set in the
complex plane
is bounded. Moreover, we investigate the associated random dynamics of
polynomials. We show that for such a polynomial semigroup, if $A$ and
$B$ are
two connected components of the Julia set, then one of $A$ and $B$
surrounds
the other. Moreover, we show that for any $n\in \Bbb{N} \cup \{\aleph_
{0}\} ,$
there exists a finitely generated polynomial semigroup with bounded
planar
postcritical set such that the cardinality of the set of all connected
components of the Julia set is equal to $n.$ Furthermore, we show
that under a
certain condition, a random Julia set is almost surely a Jordan
curve, but not
a quasicircle. Many phenomena of polynomial semigroups and random
dynamics of
polynomials that do not occur in the usual dynamics of polynomials
are found
and investigated.
http://arXiv.org/abs/math/0703591
---------------------------------------------------------------
5465. A DYNAMICAL CHARACTERIZATION OF POISSON-DIRICHLET DISTRIBUTIONS
Louis-Pierre Arguin
In this note, we show that a slight modification of a theorem of
Ruzmaikina
and Aizenman on competing particle systems on the real line leads to a
characterization of Poisson-Dirichlet distributions $PD(a,0)$.
Precisely, let $s$ be a proper random mass-partition i.e. a random
sequence
$(s_i, i\in\N)$ such that $s_1\geq s_2\geq ...$ and $\sum_i s_i=1$ a.s.
Consider ${h_i}_{i\in\N}$, an iid sequence of real random variables
with finite
Laplace transform. It is shown that if the law of $s$ is invariant
under a
random multiplicative shift $s_i e^{h_i}$ of the atoms followed by a
renormalization, then it must be a mixture of Poisson-Dirichlet
distribution
$PD(a,0)$, $a\in (0,1)$.
http://arXiv.org/abs/math/0703741
---------------------------------------------------------------
5466. GENERALIZED ZIG-ZAG PRODUCTS OF REGULAR DIGRAPHS AND BOUNDS ON
THEIR SPECTRAL EXPANSIONS
Shunichi Nomura and Akimichi Takemura
We introduce a generalization of the zig-zag product of regular digraphs
(directed graphs), which allows us to construct regular digraphs with
m ore
flexible choices of the degrees. In our generalization, we can
control the
connectivity of the resulting graph measured by its spectral
expansion. We
derive an upper bound on the spectral expansion of the generalized
zig-zag
product. Our upper bound improves on known bounds when applied to the
zig-zag
product. We also consider a special case of the generalized zig-zag
product,
where one of the components is a trivial graph whose edges are all
self-loops.
We call it a reduced zig-zag product and derive a bound on the spectral
expansion of its powers.
http://arXiv.org/abs/math/0703742
---------------------------------------------------------------
5467. IMPLICATIONS OF CONTRARIAN AND ONE-SIDED STRATEGIES FOR THE
FAIR-COIN GAME
Yasunori Horikoshi and Akimichi Takemura
We derive some results on contrarian and one-sided strategies by
Skeptic for
the fair-coin game in the framework of the game-theoretic probability
of Shafer
and Vovk \cite{sv}. In particular, concerning the rate of convergence
of the
strong law of large numbers (SLLN), we prove that Skeptic can force
that the
convergence has to be slower than or equal to $O(n^{-1/2})$. This is
achieved
by a very simple contrarian strategy of Skeptic. This type of result,
bounding
the rate of convergence from below, contrasts with more standard
results of
bounding the rate of SLLN from above by using momentum strategies. We
also
derive a corresponding one-sided result.
http://arXiv.org/abs/math/0703743
---------------------------------------------------------------
5468. ON THE INVARIANT DISTRIBUTION OF A ONE-DIMENSIONAL AVALANCHE
PROCESS
Xavier Bressaud and Nicolas Fournier
We consider an interacting particle system $(\eta_t)_{t\geq 0}$ with
values
in $\{0,1\}^{\mathbb{Z}}$, in which each vacant site becomes occupied
with rate
1, while each connected component of occupied sites become vacant
with rate
equal to its size. We show that such a process admits a unique invariant
distribution, which is exponentially mixing and can be perfectly
simulated. We
also prove that for any initial condition, the avalanche process
tends to
equilibrium exponentially fast, as time increases to infinity.
Finally, we
consider a related mean-field coagulation-fragmentation model, we
compute its
invariant distribution, and we show numerically that it is very close
to that
of the interacting particle system.
http://arXiv.org/abs/math/0703750
---------------------------------------------------------------
5469. POISSON LIMIT OF AN INHOMOGENEOUS NEARLY CRITICAL INAR(1) MODEL
L\'aszl\'o Gy\"orfi (1) and M\'arton Isp\'any (2) and Gyula Pap (2)
and Katalin Varga (1) (1)(Department of Computer Science and
Information Theory,
Budapest University of Technology and Economics) (2)(Department of
Applied
Mathematics and Probability Theory, Faculty of Informatics,
University of
Debrecen)
An inhomogeneous first--order integer--valued autoregressive (INAR(1))
process is investigated, where the autoregressive type coefficient
slowly
converges to one. It is shown that the process converges weakly to a
Poisson or
a compound Poisson distribution.
http://arXiv.org/abs/math/0703754
---------------------------------------------------------------
5470. SELF-CORRECTION OF TRANSMISSION ON REGULAR TREES
Alberto Gandolfi and Roberto Guenzani
We consider noisy binary channels on regular trees and introduce
periodic
enhancements consisting of locally self-correcting the signal in
blocks without
break of the symmetry of the model. We focus on the realistic class of
within-descent self-correction realized by identifying all
descendants $k$
generations down a vertex with their majority. We show that this also
allows
reconstruction strictly beyond the critical distortion. We further
identify the
limit at which the critical distortions of within-descent $k$ self-
corrected
transmission converge, which turns out to be the critical point for
ferromagnetic Ising model on that tree. We finally discuss how similar
phenomena take place with the biologically more plausible mechanism of
eliminating signals which are locally not coherent with the majority.
http://arXiv.org/abs/math/0703762
---------------------------------------------------------------
5471. IMPULSE CONTROL PROBLEM ON FINITE HORIZON WITH EXECUTION DELAY
Benjamin Bruder (PMA) and Huyen Pham (PMA)
We consider impulse control problems in finite horizon for diffusions
with
decision lag and execution delay. The new feature is that our general
framework
deals with the important case when several consecutive orders may be
decided
before the effective execution of the first one. This is motivated by
financial
applications in the trading of illiquid assets such as hedge funds.
We show
that the value functions for such control problems satisfy a suitable
version
of dynamic programming principle in finite dimension, which takes
into account
the past dependence of state process through the pending orders. The
corresponding Bellman partial differential equations (PDE) system is
derived,
and exhibit some peculiarities on the coupled equations, domains and
boundary
conditions. We prove a unique characterization of the value functions
to this
nonstandard PDE system by means of viscosity solutions. We then
provide an
algorithm to find the value functions and the optimal control. This
easily
implementable algorithm involves backward and forward iterations on
the domains
and the value functions, which appear in turn as original arguments
in the
proofs for the boundary conditions and uniqueness results.
http://arXiv.org/abs/math/0703769
---------------------------------------------------------------
5472. NONEQUILIBRIUM FLUCTUATIONS FOR A TAGGED PARTICLE IN MEAN-ZERO
ONE-DIMENSIONAL ZERO-RANGE PROCESSES
M.D. Jara and C. Landim and S. Sethuraman
We prove a non-equilibrium functional central limit theorem for the
position
of a tagged particle in mean-zero one-dimensional zero-range process.
The
asymptotic behavior of the tagged particle is described by a stochastic
differential equation governed by the solution of the hydrodynamic
equation.
http://arXiv.org/abs/math/0703226
---------------------------------------------------------------
5473. CENTRAL LIMIT THEOREMS FOR MULTIPLE STOCHASTIC INTEGRALS AND
MALLIAVIN CALCULUS
David Nualart and Salvador Ortiz
We give a new characterization for the convergence in distribution to a
standard normal law of a sequence of multiple stochastic integrals of
a fixed
order with variance one, in terms of the Malliavin derivatives of the
sequence.
We extend our result to the multidimensional case and prove a weak
convergence
result for a sequence of square integrable random variables.
http://arXiv.org/abs/math/0703240
---------------------------------------------------------------
5474. SAMPLE SIZE AND POSITIVE FALSE DISCOVERY RATE CONTROL FOR
MULTIPLE TESTING
Zhiyi Chi
Positive false discovery rate (pFDR) is a useful overall measure of
errors
for multiple hypothesis testing, especially when the underlying goal
is to
attain one or more discoveries. Control of pFDR critically depends on
how much
evidence is available from data to distinguish between false and true
nulls.
Oftentimes, as many aspects of the data distributions are unknown,
one may not
be able to obtain strong enough evidence from the data for pFDR
control. This
raises the question as to how much data is needed in order to attain
a target
pFDR level. We study the asymptotics of the minimum number of
observations per
null for the pFDR control associated with multiple Studentized tests
and F
tests, especially when the differences between false nulls and true
nulls are
small. For Studentized tests, we consider tests on shifts or other
parameters
associated with normal and general distributions. For F tests, we
also take
into account the effect of the number of covariates in linear
regression. The
results show that in determining the minimum sample size per null for
pFDR
control, higher order statistical properties of data are important,
and the
number of covariates is important in tests to detect regression effects.
http://arXiv.org/abs/math/0703229
---------------------------------------------------------------
5475. COUNTING MAGIC SQUARES IN QUASI-POLYNOMIAL TIME
Alexander Barvinok and Alex Samorodnitsky and and Alexander Yong
We present a randomized algorithm, which, given positive integers n
and t and
a real number 0< epsilon <1, computes the number Sigma(n, t) of n x n
non-negative integer matrices (magic squares) with the row and column
sums
equal to t within relative error epsilon. The computational
complexity of the
algorithm is polynomial in 1/epsilon and quasi-polynomial in N=nt,
that is, of
the order N^{log N}. A simplified version of the algorithm works in time
polynomial in 1/epsilon and N and estimates Sigma(n,t) within a
factor of
N^{log N}. This simplified version has been implemented. We present
results of
the implementation, state some conjectures, and discuss possible
generalizations.
http://arXiv.org/abs/math/0703227
---------------------------------------------------------------
5476. NOISE STABILITY OF FUNCTIONS WITH LOW INFLUENCES: INVARIANCE
AND OPTIMALITY II - THE NON REVERSIBLE CASE
Elchanan Mossel
We generalize an invariance principle recently obtained with
O'Donnell and
Oleszkiewicz for multilinear polynomials with low influences and bounded
degree. The generalization proven here shows invariance of the joint
distribution of several multi-linear polynomials. This in turn allows
to obtain
optimal bounds on ``noise sensitivity'' defined by non-reversible noise
operators generalizing recent results.
We present two applications of the generalized invariance
principle to the
theory of social choice. We show that Majority is asymptotically the
most
predictable function among all low influence functions given a random
sample of
the voters.
Moreover, we derive an almost tight bound in the context of Condorcet
aggregation and low influence voting schemes on a large number of
candidates.
In particular, we show that for every low influence aggregation
function, the
probability that Condorcet voting on $k$ candidates will result in a
unique
candidate that is preferable to all other is $k^{-1+o(1)}$. This
matches the
asymptotic behavior of the majority function for which the
probability is
$k^{-1-o(1)}$.
http://arXiv.org/abs/math/0703683
---------------------------------------------------------------
5477. SCHUR-WEYL DUALITY AND THE HEAT KERNEL MEASURE ON THE UNITARY
GROUP
Thierry L\'{e}vy (DMA)
We establish a convergent power series expansion for the expectation
of a
product of traces of powers of a random unitary matrix under the heat
kernel
measure. These expectations turn out to be the generating series of
certain
paths in the Cayley graph of the symmetric group. We then compute the
asymptotic distribution of a random unitary matrix under the heat kernel
measure on the unitary group $\Un$ as $N$ tends to infinity, and
prove a result
of asymptotic freeness result for independent large unitary matrices,
thus
recovering results obtained previously by Xu and Biane. We give an
interpretation of our main expansion in terms of random ramified
coverings of a
disk. Our approach is based on the Schur-Weyl duality and we extend
some of our
results to the orthogonal and symplectic cases.
http://arXiv.org/abs/math/0703690
---------------------------------------------------------------
5478. ON THE SUPREMUM OF RANDOM DIRICHLET POLYNOMIALS
Mikhail Lifshits and Michel Weber
We study the supremum of some random Dirichlet polynomials and obtain
sharp
upper and lower bounds for supremum expectation that extend the optimal
estimate of Hal\'asz-Queff\'elec and enable to cunstruct random
polynomials
with unusually small maxima.
Our approach in proving these results is entirely based on methods of
stochastic processes, in particular the metric entropy method.
http://arXiv.org/abs/math/0703691
---------------------------------------------------------------
5479. SAMPLING THE LINDEL\"OF HYPOTHESIS WITH THE CAUCHY RANDOM WALK
Mikhail Lifshits and Michel Weber
We study the behavior of the Riemann zeta function on the critical
line when
the imaginary part of the argument is sampled by the Cauchy random
walk. We
develop a complete second order theory for the corresponding system
of random
variables and show that it behaves almost like a system of non-
correlated
variables. Exploiting this fact in relation with known criteria for
almost sure
convergence allows to investigate its almost sure asymptotic behavior.
http://arXiv.org/abs/math/0703693
---------------------------------------------------------------
5480. DIVISORS OF BERNOULLI SUMS
Michel Weber
We study the asymptotic behavior of the sums of divisors when the
integers
are modelled with the Bernoulli random walk; We prealably study the
correlation
properties of the corresponding system.
http://arXiv.org/abs/math/0703696
---------------------------------------------------------------
5481. STOCHASTIC CALCULUS FOR FRACTIONAL BROWNIAN MOTION WITH HURST
EXPONENT
Jeremie Unterberger
The d-dimensional fractional Brownian motion (FBM for short)
$B_t=((B_t^{(1)},...,B_t^{(d)},t\in\R)$ with Hurst exponent $\alpha$,
$\alpha\in(0,1)$, is a d-dimensional centered, self-similar Gaussian
process
with covariance $<B_s^{(i)} B_t^{(j)}> = 1/2 \delta_{i,j}
(|s|^{2\alpha}+|t|^{2\alpha}-|t-s|^{2\alpha})$. The long-standing
problem of
defining a stochastic integration with respect to FBM (and the
related problem
of solving stochastic differential equations driven by FBM) has been
addressed
successfully by several different methods, although in each case with a
restriction on the range of either $d$ or $\alpha$. The case $\alpha=
\half$
corresponds to the usual stochastic integration with respect to Brownian
motion, while most computations become singular when $\alpha$ gets under
various threshhold values, due to the growing irregularity of the
trajectories
as $\alpha\to 0$.
We provide here a new method valid for any $d$ and for $\alpha>
{1/4}$ by
constructing an approximation $\Gamma(\eps)_t$, $\eps\to 0$, of FBM
which
allows to define iterated integrals, and then applying the geometric
rough path
theory. The approximation relies on the definition of an analytic
process
$\Gamma_z$ on the cut plane $z\in\C\setminus\R$ of which FBM appears
to be a
boundary value, and allows to understand very precisely the well-
known (see
\cite{CQ02}) but as yet a little mysterious divergence of L\'evy's
area for
$\alpha\to{1/4}$.
http://arXiv.org/abs/math/0703697
---------------------------------------------------------------
5482. EFFECTIVE MACROSCOPIC DYNAMICS OF STOCHASTIC PARTIAL
DIFFERENTIAL EQUATIONS IN PERFORATED DOMAINS
Wei Wang and Daomin Cao and Jinqiao Duan
An effective macroscopic model for a stochastic microscopic system is
derived. The original microscopic system is modeled by a stochastic
partial
differential equation defined on a domain perforated with small holes or
heterogeneities. The homogenized effective model is still a
stochastic partial
differential equation but defined on a unified domain without holes. The
solutions of the microscopic model is shown to converge to those of the
effective macroscopic model in probability distribution, as the size
of holes
diminishes to zero. Moreover, the long time effectivity of the
macroscopic
system in the sense of \emph{convergence in probability
distribution}, and the
effectivity of the macroscopic system in the sense of \emph
{convergence in
energy} are also proved.
http://arXiv.org/abs/math/0703709
---------------------------------------------------------------
5483. THE LITTLEWOOD-OFFORD PROBLEM AND INVERTIBILITY OF RANDOM MATRICES
Mark Rudelson and Roman Vershynin
We prove two basic conjectures on the distribution of the smallest
singular
value of random n times n matrices with independent entries. Under
minimal
moment assumptions, we show that the smallest singular value is of order
n^{-1/2}, which is optimal for Gaussian matrices. Moreover, we give a
optimal
estimate on the tail probability. This comes as a consequence of a
new and
essentially sharp estimate in the Littlewood-Offord problem: for
i.i.d. random
variables X_k and real numbers a_k, determine the probability P that
the sum of
a_k X_k lies near some number v. For arbitrary coefficients a_k of
the same
order of magnitude, we show that they essentially lie in an arithmetic
progression of length 1/p.
http://arXiv.org/abs/math/0703503
---------------------------------------------------------------
5484. DISPERSION MEASURE FOR SYMMETRIC, STABLE PROBABILITY DISTRIBUTIONS
Jussi I. Tyhtila
Stable distributions is an interesting and important class of
probability
distributions. They were discovered explicitly by Paul L\'{e}vy in 1925
\cite{lk}. They possess many interesting properties, most importantly
they are
by definiton invariant under addition, up to a scale. Noteworthly
they have
power-law type of decay and therefore they are an excellent model for
modelling
many natural phenomena, such as earthquakes, financial returns, and a
multitude
of social phenomena such as size distributions of cities and firms
\cite{scaling}. The major problem concerning them is that they have
an infinite
variance \cite{GK} and therefore their practical applicability is
somewhat
limited. Also they generally do not possess a density expressible in an
analytic form. This study proposes a dispersion measure for them,
drawing ideas
from Fisher information, differential geometry and most importantly, the
uncertainty principle for Fourier transform pairs \cite{Weyl}. The
study begins
with a brief discussion on characteristic functions and their
relation to
Fourier transforms and their properties, proceeds to a brief
presentation of
stable distributions and accumulates in defining a concept of
\textit{characteristic curvature}, which is proposed as a suitable
measure of
dispersion for class of stable distributions.
http://arXiv.org/abs/math/0703513
---------------------------------------------------------------
5485. PROPERTIES OF CENTERED RANDOM WALKS ON LOCALLY COMPACT GROUPS
AND LIE GROUPS
Nick Dungey
The basic aim of this paper is to study asymptotic properties of the
convolution powers K^(n) = K * K * ... * K of a possibly non-symmetric
probability density K on a locally compact, compactly generated group
G. If K
is centered, we show that the Markov operator T associated with K is
analytic
in L^p(G) for 1<p<\infty, and establish Davies-Gaffney estimates in
L^2 for the
iterated operators T^n. These results enable us to obtain various
Gaussian
bounds on K^(n). In particular, when G is a Lie group we recover and
extend
some estimates of Alexopoulos and of Varopoulos for convolution
powers of
centered densities and for the heat kernels of centered
sublaplacians. Finally,
in case G is amenable, we discover that the properties of analyticity or
Davies-Gaffney estimates hold only if K is centered.
http://arXiv.org/abs/math/0703530
---------------------------------------------------------------
5486. H{\"O}LDER CONTINUITY OF RANDOM PROCESSES
Witold Bednorz
The paper will be published in JOTP.
In the paper we prove Holder Continuity for ceratian classes of
processes
with bounded increments. The paper generalizes results obtained by
Kwapien and
Rosinski in Sample H{\"o}lder continuity of stochastic processes and
majorizing
measures. \textit{Seminar on Stochastic Analysis, Random Fields and
Applications IV, Progr. in Probab.} {\bf 58}, 155--163. Birkh{\"a}
user, Basel.
http://arXiv.org/abs/math/0703545
---------------------------------------------------------------
5487. A NOTE ON BAYESIAN NONPARAMETRIC PRIORS DERIVED FROM
EXPONENTIALLY TILTED POISSON-KINGMAN MODELS
Annalisa Cerquetti
We derive the class of normalized generalized Gamma processes from
Poisson-Kingman models (Pitman, 2003) with tempered alfa-stable mixing
distribution. Relying on this construction it can be shown that in
Bayesian
nonparametrics, results on quantities of statistical interest under
those
priors, like the analogous of the Blackwell-MacQueen prediction rules
or the
distribution of the number of distinct elements observed in a sample,
arise as
immediate consequences of Pitman's results.
http://arXiv.org/abs/math/0703552
---------------------------------------------------------------
5488. ON THE RATE OF GROWTH OF L\'EVY PROCESSES WITH NO POSITIVE
JUMPS CONDITIONED TO STAY POSITIVE
J.C. Pardo
In this article, we study the asymptotic behaviour of L\'evy
processes with
no positive jumps conditioned to stay positive. We establish integral
tests for
the lower envelope at 0 and at $+\infty$ and an analogue of
Khintchin's law of
the iterated logarithm at 0 and $+\infty$, for the upper envelope.
http://arXiv.org/abs/math/0703560
---------------------------------------------------------------
5489. HOMOGENIZED DYNAMICS OF STOCHASTIC PARTIAL DIFFERENTIAL
EQUATIONS WITH DYNAMICAL BOUNDARY CONDITIONS
Wei Wang and Jinqiao Duan
A microscopic heterogeneous system under random influence is
considered. The
randomness enters the system at physical boundary of small scale
obstacles as
well as at the interior of the physical medium. This system is
modeled by a
stochastic partial differential equation defined on a domain
perforated with
small holes (obstacles or heterogeneities), together with random
dynamical
boundary conditions on the boundaries of these small holes.
A homogenized macroscopic model for this microscopic heterogeneous
stochastic
system is derived. This homogenized effective model is a new
stochastic partial
differential equation defined on a unified domain without small
holes, with
static boundary condition only. In fact, the random dynamical boundary
conditions are homogenized out, but the impact of random forces on
the small
holes' boundaries is quantified as an extra stochastic term in the
homogenized
stochastic partial differential equation. Moreover, the validity of the
homogenized model is justified by showing that the solutions of the
microscopic
model converge to those of the effective macroscopic model in
probability
distribution, as the size of small holes diminishes to zero.
http://arXiv.org/abs/math/0703537
---------------------------------------------------------------
5490. LIMIT THEOREMS FOR RADIAL RANDOM WALKS ON PXQ-MATRICES AS P
TENDS TO INFINITY
Margit R\"osler and Michael Voit
The radial probability measures on $R^p$ are in a one-to-one
correspondence
with probability measures on $[0,\infty[$ by taking images of
measures w.r.t.
the Euclidean norm mapping. For fixed $\nu\in M^1([0,\infty[)$ and each
dimension p, we consider i.i.d. $R^p$-valued random variables
$X_1^p,X_2^p,...$
with radial laws corresponding to $\nu$ as above. We derive weak and
strong
laws of large numbers as well as a large deviation principle for the
Euclidean
length processes $S_k^p:=\|X_1^p+...+X_k^p\|$ as k,p\to\infty in
suitable ways.
In fact, we derive these results in a higher rank setting, where
$R^p$ is
replaced by the space of $p\times q$ matrices and $[0,\infty[$ by the
cone
$\Pi_q$ of positive semidefinite matrices. Proofs are based on the
fact that
the $(S_k^p)_{k\ge 0}$ form Markov chains on the cone whose transition
probabilities are given in terms Bessel functions $J_\mu$ of matrix
argument
with an index $\mu$ depending on p. The limit theorems follow from new
asymptotic results for the $J_\mu$ as $\mu\to \infty$. Similar
results are also
proven for certain Dunkl-type Bessel functions.
http://arXiv.org/abs/math/0703520
---------------------------------------------------------------
5491. KAKEYA SETS AND DIRECTIONAL MAXIMAL OPERATORS IN THE PLANE
Michael Bateman
We completely characterize the boundedness of planar directional maximal
operators on L^p. More precisely, if Omega is a set of directions, we
show that
M_Omega, the maximal operator associated to line segments in the
directions
Omega, is unbounded on L^p, for all p < infinity, precisely when
Omega admits
Kakeya-type sets. In fact, we show that if Omega does not admit
Kakeya sets,
then Omega is a generalized lacunary set, and hence M_Omega is
bounded on L^p,
for p>1.
http://arXiv.org/abs/math/0703559
---------------------------------------------------------------
5492. NONLINEAR FILTERING WITH OPTIMAL MTLL
E. Fischler and Z. Schuss
We consider the problem of nonlinear filtering of one-dimensional
diffusions
from noisy measurements. The filter is said to lose lock if the
estimation
error exits a prescribed region. In the case of phase estimation this
region is
one period of the phase measurement function, e.g., $[-\pi,\pi]$. We
show that
in the limit of small noise the causal filter that maximizes the mean
time to
loose lock is Bellman's minimum noise energy filter.
http://arXiv.org/abs/math/0703524
---------------------------------------------------------------
5493. GIBBS FRAGMENTATION TREES
Peter McCullagh and Jim Pitman and Matthias Winkel
We study fragmentation trees of Gibbs type. In the binary case, we
identify
the most general Gibbs type fragmentation tree with Aldous's beta-
splitting
model, which has an extended parameter range $\beta>-2$ with respect
to the
${\rm Beta}(\beta+1,\beta+1)$ probability distributions on which it
is based.
In the multifurcating case, we show that Gibbs fragmentation trees are
associated with the two-parameter Poisson-Dirichlet models for
exchangeable
random partitions of $\bN$, with an extended parameter range $0\le
\alpha\le 1$,
$\theta\ge -2\alpha$ and $\alpha<0$, $\theta=-m\alpha$, $m\in\bN$.
http://arxiv.org/abs/0704.0945
---------------------------------------------------------------
5494. ONE-DIMENSIONAL BROWNIAN PARTICLE SYSTEMS WITH RANK DEPENDENT
DRIFTS
Soumik Pal and Jim Pitman
We study interacting systems of linear Brownian motions whose drift
vector at
every time point is determined by the relative ranks of the coordinate
processes at that time. Our main objective has been to study the long
range
behavior of the spacings between the particles in increasing order.
For finite systems, we characterize drifts for which the spacing
system
remains stable, and show its convergence to a unique stationary joint
distribution given by independent exponential distributions with
varying means.
We also study one particular countably infinite system, where only
the minimum
Brownian particle gets a constant upward drift, and prove that
independent and
identically distributed exponential spacings remain stationary under the
dynamics of such a process.
Some related conjectures in this direction have also been discussed.
http://arxiv.org/abs/0704.0957
---------------------------------------------------------------
5495. ALMOST SURE FUNCTIONAL CENTRAL LIMIT THEOREM FOR NON-NESTLING
RANDOM WALK IN RANDOM ENVIRONMENT
Firas Rassoul-Agha and Timo Seppalainen
We consider a non-nestling random walk in a product random
environment. We
assume an exponential moment for the step of the walk, uniformly in the
environment. We prove an invariance principle (functional central limit
theorem) under almost every environment for the centered and
diffusively scaled
walk. The main point behind the invariance principle is that the
quenched mean
of the walk behaves subdiffusively.
http://arxiv.org/abs/0704.1022
---------------------------------------------------------------
5496. STATISTICS OF LOW ENERGY EXCITATIONS FOR THE DIRECTED POLYMER
IN A $1+D$ RANDOM MEDIUM ($D=1,2,3$)
Cecile Monthus and Thomas Garel
We consider a directed polymer of length $L$ in a random medium of space
dimension $d=1,2,3$. The statistics of low energy excitations as a
function of
their size $l$ is numerically evaluated. These excitations can be
divided into
bulk and boundary excitations, with respective densities $\rho^{bulk}
_L(E=0,l)$
and $\rho^{boundary}_L(E=0,l)$. We find that both densities follow
the scaling
behavior $\rho^{bulk,boundary}_L(E=0,l) = L^{-1-\theta_d}
R^{bulk,boundary}(x=l/L)$, where $\theta_d$ is the exponent governing
the
energy fluctuations at zero temperature (with the well-known exact value
$\theta_1=1/3$ in one dimension). In the limit $x=l/L \to 0$, both
scaling
functions $R^{bulk}(x)$ and $R^{boundary}(x)$ behave as $R^
{bulk,boundary}(x)
\sim x^{-1-\theta_d}$, leading to the droplet power law
$\rho^{bulk,boundary}_L(E=0,l)\sim l^{-1-\theta_d} $ in the regime $1
\ll l \ll
L$. Beyond their common singularity near $x \to 0$, the two scaling
functions
$R^{bulk,boundary}(x)$ are very different : whereas $R^{bulk}(x)$ decays
monotonically for $0<x<1$, the function $R^{boundary}(x)$ first
decays for
$0<x<x_{min}$, then grows for $x_{min}<x<1$, and finally presents a
power law
singularity $R^{boundary}(x)\sim (1-x)^{-\sigma_d}$ near $x \to 1$.
The density
of excitations of length $l=L$ accordingly decays as
$\rho^{boundary}_L(E=0,l=L)\sim L^{- \lambda_d} $ where
$\lambda_d=1+\theta_d-\sigma_d$. We obtain $\lambda_1 \simeq 0.67$, $
\lambda_2
\simeq 0.53$ and $\lambda_3 \simeq 0.39$, suggesting the possible
relation
$\lambda_d= 2 \theta_d$.
http://arXiv.org/abs/cond-mat/0602200
---------------------------------------------------------------
5497. RANDOM POLYMERS AND DELOCALIZATION TRANSITIONS
Cecile Monthus and Thomas Garel
In these proceedings, we first summarize some general properties of
phase
transitions in the presence of quenched disorder, with emphasis on the
following points: the need to distinguish typical and averaged
correlations,
the possible existence of two correlation length exponents $\nu$, the
general
bound $\nu_{FS} \geq 2/d$, the lack of self-averaging of thermodynamic
observables at criticality, the scaling properties of the
distribution of
pseudo-critical temperatures $T_c(i,L)$ over the ensemble of samples
of size
$L$. We then review our recent works on the critical properties of
various
delocalization transitions involving random polymers, namely (i) the
bidimensional wetting (ii) the Poland-Scheraga model of DNA
denaturation (iii)
the depinning transition of the selective interface model (iv) the
freezing
transition of the directed polymer in a random medium.
http://arXiv.org/abs/cond-mat/0605448
---------------------------------------------------------------
5498. NUMERICAL STUDY OF THE DIRECTED POLYMER IN A 1+3 DIMENSIONAL
RANDOM MEDIUM
Cecile Monthus and Thomas Garel
The directed polymer in a 1+3 dimensional random medium is known to
present a
disorder-induced phase transition. For a polymer of length $L$, the high
temperature phase is characterized by a diffusive behavior for the
end-point
displacement $R^2 \sim L$ and by free-energy fluctuations of order $
\Delta F(L)
\sim O(1)$. The low-temperature phase is characterized by an anomalous
wandering exponent $R^2/L \sim L^{\omega}$ and by free-energy
fluctuations of
order $\Delta F(L) \sim L^{\omega}$ where $\omega \sim 0.18$. In this
paper, we
first study the scaling behavior of various properties to localize
the critical
temperature $T_c$. Our results concerning $R^2/L$ and $\Delta F(L)$
point
towards $0.76 < T_c \leq T_2=0.79$, so our conclusion is that $T_c$
is equal or
very close to the upper bound $T_2$ derived by Derrida and coworkers
($T_2$
corresponds to the temperature above which the ratio
$\bar{Z_L^2}/(\bar{Z_L})^2$ remains finite as $L \to \infty$). We
then present
histograms for the free-energy, energy and entropy over disorder
samples. For
$T \gg T_c$, the free-energy distribution is found to be Gaussian.
For $T \ll
T_c$, the free-energy distribution coincides with the ground state
energy
distribution, in agreement with the zero-temperature fixed point
picture.
Moreover the entropy fluctuations are of order $\Delta S \sim L^{1/2}
$ and
follow a Gaussian distribution, in agreement with the droplet
predictions,
where the free-energy term $\Delta F \sim L^{\omega}$ is a near
cancellation of
energy and entropy contributions of order $L^{1/2}$.
http://arXiv.org/abs/cond-mat/0606132
---------------------------------------------------------------
5499. PROBING THE TAILS OF THE GROUND STATE ENERGY DISTRIBUTION FOR
THE DIRECTED POLYMER IN A RANDOM MEDIUM OF DIMENSION $D=1,2,3$ VIA A
MONTE-CARLO
PROCEDURE IN THE DISORDER
Cecile Monthus and Thomas Garel
In order to probe with high precision the tails of the ground-state
energy
distribution of disordered spin systems, K\"orner, Katzgraber and
Hartmann
\cite{Ko_Ka_Ha} have recently proposed an importance-sampling Monte-
Carlo
Markov chain in the disorder. In this paper, we combine their Monte-
Carlo
procedure in the disorder with exact transfer matrix calculations in
each
sample to measure the negative tail of ground state energy distribution
$P_d(E_0)$ for the directed polymer in a random medium of dimension
$d=1,2,3$.
In $d=1$, we check the validity of the algorithm by a direct
comparison with
the exact result, namely the Tracy-Widom distribution. In dimensions
$d=2$ and
$d=3$, we measure the negative tail up to ten standard deviations, which
correspond to probabilities of order $P_d(E_0) \sim 10^{-22}$. Our
results are
in agreement with Zhang's argument, stating that the negative tail
exponent
$\eta(d)$ of the asymptotic behavior $\ln P_d (E_0) \sim - | E_0 |^
{\eta(d)}$
as $E_0 \to -\infty$ is directly related to the fluctuation exponent
$\theta(d)$ (which governs the fluctuations $\Delta E_0(L) \sim L^
{\theta(d)}$
of the ground state energy $E_0$ for polymers of length $L$) via the
simple
formula $\eta(d)=1/(1-\theta(d))$. Along the paper, we comment on the
similarities and differences with spin-glasses.
http://arXiv.org/abs/cond-mat/0607411
---------------------------------------------------------------
5500. FREEZING TRANSITION OF THE RANDOM BOND RNA MODEL: STATISTICAL
PROPERTIES OF THE PAIRING WEIGHTS
Cecile Monthus and Thomas Garel
To characterize the pairing-specificity of RNA secondary structures as a
function of temperature, we analyse the statistics of the pairing
weights as
follows : for each base $(i)$ of the sequence of length N, we
consider the
$(N-1)$ pairing weights $w_i(j)$ with the other bases $(j \neq i)$ of
the
sequence. We numerically compute the probability distributions $P_1(w)
$ of the
maximal weight, the probability distribution $\Pi(Y_2)$ of the parameter
$Y_2(i)= \sum_j w_i^2(j)$, as well as the average values of the moments
$Y_k(i)= \sum_j w_i^k(j)$. We find that there are two important
temperatures
$T_c<T_{gap}$. For $T>T_{gap}$, the distribution $P_1(w)$ vanishes at
some
value $w_0(T)<1$, and accordingly the moments $\bar{Y_k(i)}$ decay
exponentially in $k$. For $T<T_{gap}$, the distributions $P_1(w)$ and
$\Pi(Y_2)$ present the characteristic Derrida-Flyvbjerg singularities at
$w,Y_2=1/n$ for $n=1,2..$. In particular, there exists a temperature-
dependent
exponent $\mu(T)$ that governs these singularities and the decay of
the moments
$ \bar{Y_k(i)} \sim 1/k^{\mu(T)}$. The exponent $\mu(T)$ grows from
$\mu(T=0)=0$ up to $\mu(T_{gap}) \sim 2$. The study of spatial
properties
indicates that the critical temperature $T_c$ where the roughness
exponent
changes from the low temperature value $\zeta \sim 0.67$ to the high
temperature value $\zeta \sim 0.5$ corresponds to the exponent $\mu
(T_c)=1$.
For $T<T_c$, there exists frozen pairs of all sizes, whereas for $T_c< T
<T_{gap}$, there exists frozen pairs, but only up to some
characteristic length
diverging as $\xi(T) \sim 1/(T_c-T)^{\nu}$ with $\nu \simeq 2$. The
similarities and differences with the weight statistics in L\'evy
sums and in
Derrida's Random Energy Model are discussed.
http://arXiv.org/abs/cond-mat/0611611
---------------------------------------------------------------
5501. DIRECTED POLYMER IN A RANDOM MEDIUM OF DIMENSION 1+3 :
MULTIFRACTAL PROPERTIES AT THE LOCALIZATION/DELOCALIZATION TRANSITION
Cecile Monthus and Thomas Garel
We consider the model of the directed polymer in a random medium of
dimension
1+3, and investigate its multifractal properties at the
localization/delocalization transition. In close analogy with models
of the
quantum Anderson localization transition, where the multifractality
of critical
wavefunctions is well established, we analyse the statistics of the
position
weights $w_L(\vec r)$ of the end-point of the polymer of length $L$
via the
moments $Y_q(L) = \sum_{\vec r} [w_L(\vec r)]^q$. We measure the
generalized
exponents $\tau(q)$ and $\tilde \tau(q)$ governing the decay of the
typical
values $Y^{typ}_q(L) = e^{\bar{\ln Y_q(L)}} \sim L^{- \tau(q)} $ and
disorder-averaged values $\bar{Y_q(L)} \sim L^{- \tilde \tau(q)} $
respectively. To understand the difference between these exponents, $
\tau(q)
\neq \tilde \tau(q)$ above some threshold $q>q_c \sim 2$, we compute the
probability distributions of $y=Y_q(L)/Y^{typ}_q(L) $ over the
samples : we
find that these distributions becomes scale invariant with a power-
law tail
$1/y^{1+x_q}$. These results thus correspond to the Ever-Mirlin
scenario [Phys.
Rev. Lett. 84, 3690 (2000)] for the statistics of Inverse
Participation Ratios
at the Anderson localization transitions. Finally, the finite-size
scaling
analysis in the critical region yields the correlation length
exponent $\nu
\sim 2$.
http://arXiv.org/abs/cond-mat/0701699
---------------------------------------------------------------
5502. DIRECTED POLYMER IN A RANDOM MEDIUM OF DIMENSION 1+1 AND 1+3:
WEIGHTS STATISTICS IN THE LOW-TEMPERATURE PHASE
Cecile Monthus and Thomas Garel
We consider the low-temperature $T<T_c$ disorder-dominated phase of the
directed polymer in a random potentiel in dimension 1+1 (where $T_c=
\infty$)
and 1+3 (where $T_c<\infty$). To characterize the localization
properties of
the polymer of length $L$, we analyse the statistics of the weights
$w_L(\vec
r)$ of the last monomer as follows. We numerically compute the
probability
distributions $P_1(w)$ of the maximal weight $w_L^{max}= max_{\vec r}
[w_L(\vec
r)]$, the probability distribution $\Pi(Y_2)$ of the parameter $Y_2(L)=
\sum_{\vec r} w_L^2(\vec r) $ as well as the average values of the
higher order
moments $Y_k(L)= \sum_{\vec r} w_L^k(\vec r)$. We find that there
exists a
temperature $T_{gap}<T_c$ such that (i) for $T<T_{gap}$, the
distributions
$P_1(w)$ and $\Pi(Y_2)$ present the characteristic Derrida-Flyvbjerg
singularities at $w=1/n$ and $Y_2=1/n$ for $n=1,2..$. In particular,
there
exists a temperature-dependent exponent $\mu(T)$ that governs the main
singularities $P_1(w) \sim (1-w)^{\mu(T)-1}$ and $\Pi(Y_2) \sim
(1-Y_2)^{\mu(T)-1}$ as well as the power-law decay of the moments $
\bar{Y_k(i)} \sim 1/k^{\mu(T)}$. The exponent $\mu(T)$ grows from the
value
$\mu(T=0)=0$ up to $\mu(T_{gap}) \sim 2$. (ii) for $T_{gap}<T<T_c$, the
distribution $P_1(w)$ vanishes at some value $w_0(T)<1$, and
accordingly the
moments $\bar{Y_k(i)}$ decay exponentially as $(w_0(T))^k$ in $k$. The
histograms of spatial correlations also display Derrida-Flyvbjerg
singularities
for $T<T_{gap}$. Both below and above $T_{gap}$, the study of typical
and
averaged correlations is in full agreement with the droplet scaling
theory.
http://arXiv.org/abs/cond-mat/0702131
---------------------------------------------------------------
5503. ON THE CRITICAL WEIGHT STATISTICS OF THE RANDOM ENERGY MODEL
AND OF THE DIRECTED POLYMER ON THE CAYLEY TREE
Cecile Monthus and Thomas Garel
We consider the critical point of two mean-field disordered models :
(i) the
Random Energy Model (REM), introduced by Derrida as a mean-field spin-
glass
model of $N$ spins (ii) the Directed Polymer of length $N$ on a
Cayley Tree
(DPCT) with random bond energies. Both models are known to exhibit a
freezing
transition between a high temperature phase where the entropy is
extensive and
a low-temperature phase of finite entropy. In this paper, we study
the weight
statistics at criticality via the entropy $S=-\sum w_i \ln w_i$ and the
generalized moments $Y_k=\sum w_i^k$, where the $w_i$ are the
Boltzmann weights
of the $2^N$ configurations. In the REM, we find that the critical
weight
statistics is governed by the finite-size exponent $\nu=2$ : the
entropy scales
as $\bar{S}_N(T_c) \sim N^{1/2}$, the typical values $e^{\bar{\ln
Y_k}}$ decay
as $N^{-k/2}$, and the disorder-averaged values $\bar{Y_k}$ are
governed by
rare events and decay as $N^{-1/2}$ for any $k>1$. For the DPCT, we
find that
the entropy scales similarly as $\bar{S}_N(T_c) \sim N^{1/2}$,
whereas another
exponent $\nu'=1$ governs the $Y_k$ statistics : the typical values
$e^{\bar{\ln Y_k}}$ decay as $N^{-k}$, the disorder-averaged values $
\bar{Y_k}$
decay as $N^{-1}$ for any $k>1$. As a consequence, the asymptotic
probability
distribution $\bar{\pi}_{N=\infty}(q)$ of the overlap $q$, beside the
delta
function $\delta(q)$ which bears the whole normalization, contains an
isolated
point at $q=1$, as a memory of the delta peak $(1-T/T_c) \delta(q-1)$
of the
low-temperature phase $T<T_c$. The associated value $\bar{\pi}_{N=
\infty}(q=1)$
is finite for the DPCT, and diverges as $\bar{\pi}_{N=\infty}(q=1) \sim
N^{1/2}$ for the REM.
http://arXiv.org/abs/cond-mat/0703017
---------------------------------------------------------------
5504. AN ISOPERIMETRIC INEQUALITY FOR UNIFORMLY LOG-CONCAVE MEASURES
AND UNIFORMLY CONVEX BODIES
Emanuel Milman and Sasha Sodin
We prove an isoperimetric inequality for uniformly log-concave
measures and
for the uniform measure on a uniformly convex body. These
inequalities imply
the log-Sobolev inequalities proved by Bobkov and Ledoux and Bobkov and
Zegarlinski. We also recover a concentration inequality for uniformly
convex
bodies, similar to that proved by Gromov and Milman.
http://arXiv.org/abs/math/0703857
---------------------------------------------------------------
5505. LONG RANGE PERCOLATION MIXING TIME
Itai Benjamini and Noam Berger and Ariel Yadin
We provide an estimate, sharp up to poly-logarithmic factors, of the
asymptotically almost sure mixing time of the graph created by long-
range
percolation on the cycle of length N (Z/NZ). While it is known that
the almost
sure diameter drops from linear to poly-logarithmic as the exponent s
decreases
below 2, the almost sure mixing time drops from N^2 only to N^(s-1)
(up to
poly-logarithmic factors).
http://arXiv.org/abs/math/0703872
---------------------------------------------------------------
5506. COALESCENT PROCESSES ARISING IN A STUDY OF DIFFUSIVE CLUSTERING
Andreas Greven and Vlada Limic and Anita Winter
This paper studies spatial coalescents on $\Z^2$. In our setting, the
partition elements are located at the sites of $\Z^2$ and undergo
local delayed
coalescence and migration. The system starts in either locally finite
configurations or in configurations containing countably many partition
elements per site.
Our goal is to determine the longtime behavior with an initial
population of
countably many individuals per site restricted to a box $[-t^{\alpha/2},
t^{\alpha/2}]^2 \cap \Z^2$ and observed at time $t^\beta$ with $1
\geq \beta
\geq \alpha\ge 0$. We study both asymptotics, as $t\to\infty$, for a
fixed
value of $\alpha$ as the parameter $\beta\in[\alpha,1]$ varies, and
for a fixed
$\beta=1$, as the parameter $\alpha\in [0,1]$ varies.
A new random object, the so-called {\em coalescent with rebirth}, is
constructed and shown to arise in the limit. In view of future
applications we
introduce the spatial coalescent with rebirth and study its longtime
asymptotics as well. The present paper is the basis for forthcoming
work, where
the genealogies in interacting Moran models and Fisher-Wright
diffusions on
$\Z^2$ are studied. There the coalescent with rebirth is needed to
describe the
``complete'' genealogical forests, i.e., the genealogical structures
which
include also the ``fossils''.
http://arXiv.org/abs/math/0703875
---------------------------------------------------------------
5507. ON BOUNDED SOLUTIONS OF THE BALANCED GENERALIZED PANTOGRAPH
EQUATION
Leonid Bogachev and Gregory Derfel and Stanislav Molchanov and and
John Ockendon
The question about the existence and characterization of bounded
solutions to
linear functional-differential equations with both advanced and delayed
arguments was posed in early 1970s by T. Kato in connection with the
analysis
of the pantograph equation, y'(x)=ay(qx)+by(x). In the present paper,
we answer
this question for the balanced generalized pantograph equation of the
form -a_2
y''(x)+a_1 y'(x)+y(x)=int_0^infty y(qx) m(dq), where a_1 > or = 0,
a_2 > or =
0, a_1^2+a_2^2>0, and m is a probability measure. Namely, setting
K:=int_0^infty ln(q) m(dq), we prove that if K < or = 0 then the
equation does
not have nontrivial (i.e., nonconstant) bounded solutions, while if
K>0 then
such a solution exists. The result in the critical case, K=0, settles a
long-standing problem. The proof exploits the link with the theory of
Markov
processes, in that any solution of the balanced pantograph equation
is an
L-harmonic function relative to the generator L of a certain
diffusion process
with "multiplication" jumps. The paper also includes three
"elementary" proofs
for the simple prototype equation y'(x)+y(x)=(1/2)y(qx)+(1/2)y(x/q),
based on
perturbation, analytical, and probabilistic techniques, respectively,
which may
appear useful in other situations as efficient exploratory tools.
http://arXiv.org/abs/math/0703897
---------------------------------------------------------------
5508. ELECTRIC CURRENTS IN INFINITE NETWORKS
Peter G. Doyle
In this survey, we present the basic facts about conduction in infinite
networks. This survey is based on the work of Flanders, Zemanian, and
Thomassen, who developed the theory of infinite networks from
scratch. Here we
show how to get a more complete theory by paralleling the well-
developed theory
of conduction on open Riemann surfaces. Like Flanders and Thomassen,
we take as
a test case for the theory the problem of determining the resistance
across an
edge of a d-dimensional grid of 1-ohm resistors.
http://arXiv.org/abs/math/0703899
---------------------------------------------------------------
5509. PARAMETRIC ESTIMATION FOR PLANAR RANDOM FLIGHTS OBSERVED AT
DISCRETE TIMES
Alessandro De Gregorio
We deal with a planar random flight $\{(X(t),Y(t)),0<t\leq T\}$
observed at
$n+1$ equidistant times $t_i=i\Delta_n,i=0,1,...,n$. The aim of this
paper is
to estimate the unknown value of the parameter $\lambda$, the
underlying rate
of the Poisson process. The planar random flights are not markovian,
then we
use an alternative argument to derive a pseudo-maximum likelihood
estimator
$\hat{\lambda}$ of the parameter $\lambda$. We consider two different
types of
asymptotic schemes and show the consistency, the asymptotic normality
and
efficiency of the estimator proposed. A Monte Carlo analysis for
small sample
size $n$ permits us to analyze the empirical performance of $\hat
{\lambda}$.
A different approach permits us to introduce an alternative
estimator of
$\lambda$ which is consistent, asymptotically normal and asymptotically
efficient without the request of other assumptions.
http://arXiv.org/abs/math/0703887
---------------------------------------------------------------
5510. OPTIMAL DEFERRED LIFE ANNUITIES TO MINIMIZE THE PROBABILITY OF
LIFETIME RUIN
Erhan Bayraktar and Virginia R. Young
We find the minimum probability of lifetime ruin of an investor who can
invest in a market with a risky and a riskless asset and can purchase a
deferred annuity. Although we let the admissible set of strategies of
annuity
purchasing process to be increasing adapted processes, we find that the
individual will not buy a deferred life annuity unless she can cover
all her
consumption via the annuity and have enough wealth left over to
sustain her
until the end of the deferral period.
http://arXiv.org/abs/math/0703862
---------------------------------------------------------------
5511. THE BIHAM-MIDDLETON-LEVINE TRAFFIC MODEL FOR A SINGLE JUNCTION
Itai Benjamini and Ori Gurel-Gurevich and Roey Izkovsky
In the Biham-Middleton-Levine traffic model cars are placed in some
density p
on a two dimensional torus, and move according to a (simple) set of
predefined
rules. Computer simulations show this system exhibits many interesting
phenomena: for low densities the system self organizes such that cars
flow
freely while for densities higher than some critical density the
system gets
stuck in an endless traffic jam. However, apart from the simulation
results
very few properties of the system were proven rigorously to date. We
introduce
a simplified version of this model in which cars are placed in a
single row and
column (a junction) and show that similar phenomena of self-
organization of the
system and phase transition still occur.
http://arXiv.org/abs/math/0703201
---------------------------------------------------------------
5512. STABILITY IN RANDOM BOOLEAN CELLULAR AUTOMATA ON THE INTEGER
LATTICE
F.M.Dekking and L. van Driel
We consider random boolean cellular automata on the integer lattice,
i.e.,
the cells are identified with the integers from 1 to $N$. The
behaviour of the
automaton is mainly determined by the support of the random variable
that
selects one of the sixteen possible Boolean rules, independently for
each cell.
A cell is said to stabilize if it will not change its state anymore
after some
time. We classify the random boolean automata according to the
positivity of
their probability of stabilization. Here is an example of a
consequence of our
results: if the support contains at least 5 rules, then
asymptotically as $N$
tends to infinity the probability of stabilization is positive,
whereas there
exist random boolean cellular automata with 4 rules in their support
for which
this probability tends to 0.
http://arxiv.org/abs/0704.2183
---------------------------------------------------------------
5513. FACTOR ANALYSIS AND ALTERNATING MINIMIZATION
Lorenzo Finesso and Peter Spreij
In this paper we make a first attempt at understanding how to build an
optimal approximate normal factor analysis model. The criterion we
have chosen
to evaluate the distance between different models is the I-divergence
between
the corresponding normal laws. The algorithm that we propose for the
construction of the best approximation is of an the alternating
minimization
kind.
http://arxiv.org/abs/0704.2208
---------------------------------------------------------------
5514. ON THE COMPUTATIONAL COMPLEXITY OF MCMC-BASED ESTIMATORS IN
LARGE SAMPLES
Alexandre Belloni and Victor Chernozhukov
In this paper we examine the implications of the statistical large
sample
theory for the computational complexity of Bayesian and quasi-Bayesian
estimation carried out using Metropolis random walks. Our analysis is
motivated
by the Laplace-Bernstein-Von Mises central limit theorem, which
states that in
large samples the posterior or quasi-posterior approaches a normal
density.
Using this observation, we establish polynomial bounds on the
computational
complexity of general Metropolis random walks methods in large
samples. Our
analysis covers cases, where the underlying log-likelihood or extremum
criterion function is possibly non-concave, discontinuous, and of
increasing
dimension. However, the central limit theorem restricts the
deviations from
continuity and log-concavity of the log-likelihood or extremum criterion
function in a very specific manner. Under minimal assumptions for the
central
limit theorem framework to hold, we show that the Metropolis
algorithm is
theoretically efficient even for the canonical Gaussian walk which is
studied
in detail. Specifically, we show that the running time of the
algorithm in
large samples is bounded in probability by a polynomial in the parameter
dimension d, and, in particular, is of stochastic order d^2 in the
leading
cases after the burn-in period. We then give an application to
exponential and
curved exponential families of increasing dimension.
http://arxiv.org/abs/0704.2167
---------------------------------------------------------------
5515. PROBABILIT\'ES ET FLUCTUATIONS QUANTIQUES (PROBABILITIES AND
QUANTUM FLUCTUATIONS)
Michel Fliess (LIX and Inria Futurs)
This note is sketching a simple and natural mathematical construction
for
explaining the probabilistic nature of quantum mechanics. It employs
nonstandard analysis and is based on Feynman's interpretation of the
Heisenberg
uncertainty principle, i.e., of the quantum fluctuations, which was
brought to
the forefront in some fractal approaches. It results, as in Nelson's
stochastic
mechanics, in stochastic differential equations which are deduced from
infinitesimal random walks. An extended english abstract gives most
of the
details.
http://arxiv.org/abs/0704.2019
---------------------------------------------------------------
5516. NEIGHBORING CLUSTERS IN BERNOULLI PERCOLATION
Ad\'{a}m Tim\'{a}r
We consider Bernoulli percolation on a locally finite quasi-transitive
unimodular graph and prove that two infinite clusters cannot have
infinitely
many pairs of vertices at distance 1 from one another or, in other
words, that
such graphs exhibit ``cluster repulsion.'' This partially answers a
question of
H\"{a}ggstr\"{o}m, Peres and Schonmann.
http://arXiv.org/abs/math/0702873
---------------------------------------------------------------
5517. PERCOLATION ON NONUNIMODULAR TRANSITIVE GRAPHS
\'{A}d\'{a}m Tim\'{a}r
We extend some of the fundamental results about percolation on
unimodular
nonamenable graphs to nonunimodular graphs. We show that they cannot
have
infinitely many infinite clusters at critical Bernoulli percolation.
In the
case of heavy clusters, this result has already been established, but
it also
follows from one of our results. We give a general necessary
condition for
nonunimodular graphs to have a phase with infinitely many heavy
clusters. We
present an invariant spanning tree with $p_c=1$ on some nonunimodular
graph.
Such trees cannot exist for nonamenable unimodular graphs. We show a
new way of
constructing nonunimodular graphs that have properties more peculiar
than the
ones previously known.
http://arXiv.org/abs/math/0702875
---------------------------------------------------------------
5518. LOWER BOUNDS FOR THE DENSITY OF LOCALLY ELLIPTIC IT\^{O} PROCESSES
Vlad Bally
We give lower bounds for the density $p_T(x,y)$ of the law of $X_t$, the
solution of $dX_t=\sigma (X_t) dB_t+b(X_t) dt,X_0=x,$ under the
following local
ellipticity hypothesis: there exists a deterministic differentiable
curve $x_t,
0\leq t\leq T$, such that $x_0=x, x_T=y$ and $\sigma \sigma ^*(x_t)>0,
$ for all
$t\in \lbrack 0,T].$ The lower bound is expressed in terms of a distance
related to the skeleton of the diffusion process. This distance
appears when we
optimize over all the curves which verify the above ellipticity
assumption. The
arguments which lead to the above result work in a general context which
includes a large class of Wiener functionals, for example, It\^{o}
processes.
Our starting point is work of Kohatsu-Higa which presents a general
framework
including stochastic PDE's.
http://arXiv.org/abs/math/0702879
---------------------------------------------------------------
5519. WAITING FOR REGULATORY SEQUENCES TO APPEAR
Richard Durrett and Deena Schmidt
One possible explanation for the substantial organismal differences
between
humans and chimpanzees is that there have been changes in gene
regulation.
Given what is known about transcription factor binding sites, this
motivates
the following probability question: given a 1000 nucleotide region in
our
genome, how long does it take for a specified six to nine letter word
to appear
in that region in some individual? Stone and Wray [Mol. Biol. Evol.
18 (2001)
1764--1770] computed 5,950 years as the answer for six letter words.
Here, we
will show that for words of length 6, the average waiting time is
100,000
years, while for words of length 8, the waiting time has mean 375,000
years
when there is a 7 out of 8 letter match in the population consensus
sequence
(an event of probability roughly 5/16) and has mean 650 million years
when
there is not. Fortunately, in biological reality, the match to the
target word
does not have to be perfect for binding to occur. If we model this by
saying
that a 7 out of 8 letter match is good enough, the mean reduces to
about 60,000
years.
http://arXiv.org/abs/math/0702883
---------------------------------------------------------------
5520. INTEGRATION BY PARTS FORMULA FOR LOCALLY SMOOTH LAWS AND
APPLICATIONS TO SENSITIVITY COMPUTATIONS
Vlad Bally and Marie-Pierre Bavouzet and Marouen Messaoud
We consider random variables of the form $F=f(V_1,...,V_n)$, where $f
$ is a
smooth function and $V_i,i\in\mathbb{N}$, are random variables with
absolutely
continuous law $p_i(y) dy$. We assume that $p_i$, $i=1,...,n$, are
piecewise
differentiable and we develop a differential calculus of Malliavin
type based
on $\partial\ln p_i$. This allows us to establish an integration by
parts
formula $E(\partial_i\phi(F)G)=E(\phi(F)H_i(F,G))$, where $H_i(F,G)$
is a
random variable constructed using the differential operators acting
on $F$ and
$G.$ We use this formula in order to give numerical algorithms for
sensitivity
computations in a model driven by a L\'{e}vy process.
http://arXiv.org/abs/math/0702884
---------------------------------------------------------------
5521. A FLEMING--VIOT PROCESS AND BAYESIAN NONPARAMETRICS
Stephen G. Walker and Spyridon J. Hatjispyros and Theodoros Nicoleris
This paper provides a construction of a Fleming--Viot measure valued
diffusion process, for which the transition function is known, by
extending
recent ideas of the Gibbs sampler based Markov processes. In
particular, we
concentrate on the Chapman--Kolmogorov consistency conditions which
allows a
simple derivation of such a Fleming--Viot process, once a key and
apparently
new combinatorial result for P\'{o}lya-urn sequences has been
established.
http://arXiv.org/abs/math/0702885
---------------------------------------------------------------
5522. ON THE SIGNAL-TO-INTERFERENCE RATIO OF CDMA SYSTEMS IN
WIRELESS COMMUNICATIONS
Z. D. Bai and Jack W. Silverstein
Let $\{s_{ij}:i,j=1,2,...\}$ consist of i.i.d. random variables in
$\mathbb{C}$ with $\mathsf{E}s_{11}=0$, $\mathsf{E}|s_{11}|^2=1$. For
each
positive integer $N$, let
$\mathbf{s}_k={\mathbf{s}}_k(N)=(s_{1k},s_{2k},...,s_{Nk})^T$, $1\leq
k\leq K$,
with $K=K(N)$ and $K/N\to c>0$ as $N\to\infty$. Assume for fixed
positive
integer $L$, for each $N$ and $k\leq K$,
${\bolds\alpha}_k=(\alpha_k(1),...,\alpha_k(L))^T$ is random,
independent of
the $s_{ij}$, and the empirical distribution of $(\alpha_1,...,
\alpha_K)$, with
probability one converging weakly to a probability distribution $H$ on
$\mathbb{C}^L$. Let ${\bolds\beta
}_k={\bolds\beta}_k(N)=(\alpha_k(1)\mathbf{s}_k^T,...,\alpha_k(L)\m
athbf{s}_k^T)^T$ and set $C=C(N)=(1/N)\sum_{k=2}^K{\bolds \beta}_k
{\bolds
\beta}_k^*$. Let $\sigma^2>0$ be arbitrary. Then define
$SIR_1=(1/N){\bolds\beta}^*_1(C+\sigma^2I)^{-1}{\bolds\beta}_1$, which
represents the best signal-to-interference ratio for user 1 with
respect to the
other $K-1$ users in a direct-sequence code-division multiple-access
system in
wireless communications. In this paper it is proven that, with
probability 1,
$SIR_1$ tends, as $N\to\infty$, to the limit
$\sum_{\ell,\ell'=1}^L\bar{\alpha}_1(\ell) alpha_1(\ell')a_{\ell,
\ell'},$ where
$A=(a_{\ell,\ell'})$ is nonrandom, Hermitian positive definite, and
is the
unique matrix of such type satisfying $A=\bigl(c
\mathsf{E}\frac{{\bolds\alpha}{\bolds
\alpha}^*}{1+{\bolds\alpha}^*A{\bolds\alpha}}+\sigma^2I_L\bigr)^{-1}
$, where
${\bolds\alpha}\in \mathbb{C}^L$ has distribution $H$. The result
generalizes
those previously derived under more restricted assumptions.
http://arXiv.org/abs/math/0702888
---------------------------------------------------------------
5523. ON THE OPTIMAL DIVIDEND PROBLEM FOR A SPECTRALLY NEGATIVE L\'{E}
VY PROCESS
Florin Avram and Zbigniew Palmowski and Martijn R. Pistorius
In this paper we consider the optimal dividend problem for an insurance
company whose risk process evolves as a spectrally negative L\'{e}vy
process in
the absence of dividend payments. The classical dividend problem for an
insurance company consists in finding a dividend payment policy that
maximizes
the total expected discounted dividends. Related is the problem where
we impose
the restriction that ruin be prevented: the beneficiaries of the
dividends must
then keep the insurance company solvent by bail-out loans. Drawing on
the
fluctuation theory of spectrally negative L\'{e}vy processes we give an
explicit analytical description of the optimal strategy in the set of
barrier
strategies and the corresponding value function, for either of the
problems.
Subsequently we investigate when the dividend policy that is optimal
among all
admissible ones takes the form of a barrier strategy.
http://arXiv.org/abs/math/0702893
---------------------------------------------------------------
5524. RETRIEVING CONVEX BODIES FROM RESTRICTED COVARIOGRAM FUNCTIONS
Gennadiy Averkov (University of Magdeburg) and Gabriele Bianchi
(Universita` di Firenze)
The covariogram g_K(x) of a convex body K \subseteq E^d is the
function which
associates to each x \in E^d the volume of the intersection of K with
K+x.
Matheron asked whether g_K determines K, up to translations and
reflections in
a point. Positive answers to Matheron's question have been obtained
for large
classes of planar convex bodies, while for d\geq 3 there are both
positive and
negative results.
One of the purposes of this paper is to sharpen some of the known
results on
Matheron's conjecture indicating how much of the covariogram
information is
needed to get the uniqueness of determination. We indicate some
subsets of the
support of the covariogram, with arbitrarily small Lebesgue measure,
such that
the covariogram, restricted to those subsets, identifies certain
geometric
properties of the body. These results are more precise in the planar
case, but
some of them, both positive and negative ones, are proved for bodies
of any
dimension. Moreover some results regard most convex bodies, in the Baire
category sense. Another purpose is to extend the class of convex
bodies for
which Matheron's conjecture is confirmed by including all planar
convex bodies
possessing two non-degenerate boundary arcs being reflections of each
other.
http://arXiv.org/abs/math/0702892
---------------------------------------------------------------
5525. GAUSSIAN FLUCTUATIONS FOR \BETA ENSEMBLES
Rowan Killip
We study the Circular and Jacobi $\beta$-Ensembles and prove Gaussian
fluctuations for the number of points in one or more intervals in the
macroscopic scaling limit.
http://arXiv.org/abs/math/0703140
---------------------------------------------------------------
5526. FRONT PROPAGATION IN AN EXCLUSION ONE-DIMENSIONAL REACTIVE
DYNAMICS
Milton Jara and Gregorio Moreno and Alejandro F. Ramirez
We consider an exclusion process representing a reactive dynamics of
a pulled
front on the integer lattice, describing the dynamics of first class $X$
particles moving as a simple symmetric exclusion process, and static
second
class $Y$ particles. When an $X$ particle jumps to a site with a $Y$
particle,
their position is intechanged and the $Y$ particle becomes an $X$ one.
Initially, there is an arbitrary configuration of $X$ particles at
sites $...,
-1,0$, and $Y$ particles only at sites $1,2,...$, with a product
Bernoulli law
of parameter $\rho,0<\rho<1$. We prove a law of large numbers and a
central
limit theorem for the front defined by the right-most visited site of
the $X$
particles at time $t$. These results corroborate Monte-Carlo simulations
performed in a similar context. We also prove that the law of the $X$
particles
as seen from the front converges to a unique invariant measure. The
proofs use
regeneration times: we present a direct way to define them within
this context.
http://arXiv.org/abs/math/0703173
---------------------------------------------------------------
5527. SCALING LIMIT FOR A CLASS OF GRADIENT FIELDS WITH NON-CONVEX
POTENTIALS
Marek Biskup and Herbert Spohn
We consider gradient fields $(\phi_x\colon x\in\Z^d)$ whose law takes
the
Gibbs-Boltzmann form $Z^{-1}\exp\{-\sum_{< x,y>}V(\phi_y-\phi_x)\}$
where the
sum runs over nearest neighbors. We assume that $V$ admits the
representation
$$ V(\eta)= - \log\int\varrho(\textd\kappa) \exp
\bigl[-\tfrac{1}{2}\kappa\eta^2\bigr] $$ where $\varrho$ is a
positive measure
with compact support in $(0,\infty)$. Hence $V$ is symmetric and non-
convex in
general. While for strictly convex $V$'s the translation-invariant,
ergodic
gradient Gibbs measures are completely characterized by their tilt, a
non-convex potential as above may lead to several ergodic gradient Gibbs
measures with zero tilt. Still, every ergodic, zero-tilt gradient
Gibbs measure
for the potential $V$ from above scales to a Gaussian free field.
http://arxiv.org/abs/0704.3086
---------------------------------------------------------------
5528. THE CONTOUR OF SPLITTING TREES IS A L\'EVY PROCESS
Amaury Lambert (FESE)
Splitting trees are those random trees where individuals give birth at
constant rate during a lifetime with general distribution, to i.i.d.
copies of
themselves. The width process of a splitting tree is then a binary,
homogeneous
Crump--Mode--Jagers (CMJ) process, and is not Markovian unless the
lifetime
distribution is exponential. Here, we allow the birth rate to be
infinite, that
is, pairs of birth times and lifespans of newborns form a Poisson
point process
along the lifetime of their mother, with possibly infinite intensity
measure. A
splitting tree is a random (so-called) chronological tree. Each
element of a
chronological tree is a (so-called) existence point $(v,\tau)$ of some
individual $v$ (vertex) in a discrete tree, where $\tau$ is a
nonnegative real
number called chronological level (time). We introduce a total order on
existence points, called linear order, and a mapping $\phi$ from the
tree into
the real line which preserves this order. The inverse of $\phi$ is
called the
exploration process, and the projection of this inverse on
chronological levels
the contour process. For splitting trees truncated up to level $\tau
$, we prove
that thus defined contour process is a L\'evy process reflected below
$\tau$
and killed upon hitting 0. This allows to derive properties of (i)
splitting
trees: conceptual proof of Le Gall--Le Jan's theorem in the finite
variation
case, exceptional points, coalescent point process, age distribution;
(ii) CMJ
processes: one-dimensional marginals, conditionings, limit theorems,
asymptotic
numbers of individuals with infinite vs finite descendances.
http://arxiv.org/abs/0704.3098
---------------------------------------------------------------
5529. TWO-PARAMETER POISSON-DIRICHLET MEASURES AND REVERSIBLE
EXCHANGEABLE FRAGMENTATION-COALESCENCE PROCESSES
Jean Bertoin (PMA and Dma)
We show that for $0<\alpha<1$ and $\theta>-\alpha$, the Poisson-
Dirichlet
distribution with parameter $(\alpha, \theta)$ is the unique reversible
distribution of a rather natural fragmentation-coalescence process. This
completes earlier results in the literature for certain split and merge
transformations and the parameter $\alpha =0$.
http://arxiv.org/abs/0704.3122
---------------------------------------------------------------
5530. HOW TO CLEAN A DIRTY FLOOR: PROBABILISTIC POTENTIAL THEORY AND
THE DOBRUSHIN UNIQUENESS THEOREM
Thierry de la Rue and Roberto Fernandez and Alan D. Sokal
Motivated by the Dobrushin uniqueness theorem in statistical
mechanics, we
consider the following situation: Let \alpha be a nonnegative matrix
over a
finite or countably infinite index set X, and define the "cleaning
operators"
\beta_h = I_{1-h} + I_h \alpha for h: X \to [0,1] (here I_f denotes the
diagonal matrix with entries f). We ask: For which "cleaning
sequences" h_1,
h_2, ... do we have c \beta_{h_1} ... \beta_{h_n} \to 0 for a
suitable class of
"dirt vectors" c? We show, under a modest condition on \alpha, that
this occurs
whenever \sum_i h_i = \infty everywhere on X. More generally, we
analyze the
cleaning of subsets \Lambda \subseteq X and the final distribution of
dirt on
the complement of \Lambda. We show that when supp(h_i) \subseteq
\Lambda with
\sum_i h_i = \infty everywhere on \Lambda, the operators \beta_{h_1} ...
\beta_{h_n} converge as n \to \infty to the "balayage operator" \Pi_
\Lambda =
\sum_{k=0}^\infty (I_\Lambda \alpha)^k I_{\Lambda^c). These results are
obtained in two ways: by a fairly simple matrix formalism, and by a more
powerful tree formalism that corresponds to working with formal power
series in
which the matrix elements of \alpha are treated as noncommuting
indeterminates.
http://arxiv.org/abs/0704.3156
---------------------------------------------------------------
5531. MULTIPLE PATTERN MATCHING: A MARKOV CHAIN APPROACH
Manuel Lladser and Meredith D. Betterton and Rob Knight
RNA motifs typically consist of short, modular patterns that include
base
pairs formed within and between modules. Estimating the abundance of
these
patterns is of fundamental importance for assessing the statistical
significance of matches in genomewide searches, and for predicting
whether a
given function has evolved many times in different species or arose
from a
single common ancestor. In this manuscript, we review in an
integrated and
self-contained manner some basic concepts of automata theory, generating
functions and transfer matrix methods that are relevant to pattern
analysis in
biological sequences. We formalize, in a general framework, the
concept of
Markov chain embedding to analyze patterns in random strings produced
by a
memoryless source. This conceptualization, together with the
capability of
automata to recognize complicated patterns, allows a systematic
analysis of
problems related to the occurrence and frequency of patterns in
random strings.
The applications we present focus on the concept of synchronization of
automata, as well as automata used to search for a finite number of
keywords
(including sets of patterns generated according to base pairing
rules) in a
general text.
http://arxiv.org/abs/0704.3221
---------------------------------------------------------------
5532. ENTANGLEMENT IN THE QUANTUM ISING MODEL
Geoffrey Grimmett and Tobias Osborne and Petra Scudo
We study the asymptotic scaling of the entanglement of a block of
spins for
the ground state of the one-dimensional quantum Ising model with
transverse
field. When the field is sufficiently strong, the entanglement grows
at most
logarithmically in the number of spins. The proof utilises a
transformation to
a model of classical probability called the continuum random-cluster
model, and
is based on a property of the latter model termed ratio weak-mixing.
Our proof
applies equally to a large class of disordered interactions.
http://arxiv.org/abs/0704.2981
---------------------------------------------------------------
5533. GIBBS MEASURES ON BROWNIAN CURRENTS
Massimiliano Gubinelli and Jozsef Lorinczi
Motivated by applications to quantum field theory we consider Gibbs
measures
for which the reference measure is Wiener measure and the interaction
is given
by a double stochastic integral and a pinning external potential. In
order
properly to characterize these measures through DLR equations, we are
led to
lift Wiener measure and other objects to a space of configurations
where the
basic observables are not only the position of the particle at all
times but
also the work done by test vector fields. We prove existence and basic
properties of such Gibbs measures in the small coupling regime by
means of
cluster expansion.
http://arxiv.org/abs/0704.3237
---------------------------------------------------------------
5534. HYDRODYNAMIC LIMIT FOR A PARTICLE SYSTEM WITH DEGENERATE RATES
Patricia Goncalves and Claudio Landim and Cristina Toninelli
We study the hydrodynamic limit for some conservative particle
systems with
degenerate rates, namely with nearest neighbor exchange rates which
vanish for
certain configurations. These models belong to the class of {\sl
kinetically
constrained lattice gases} (KCLG) which have been introduced and
intensively
studied in physics literature as simple models for the liquid/glass
transition.
Due to the degeneracy of rates for KCLG there exists {\sl blocked
configurations} which do not evolve under the dynamics and in general
the
hyperplanes of configurations with a fixed number of particles can be
decomposed into different irreducible sets. As a consequence, both
the Entropy
and Relative Entropy method cannot be straightforwardly applied to
prove the
hydrodynamic limit. In particular, some care should be put when
proving the One
and Two block Lemmas which guarantee local convergence to
equilibrium. We show
that, for initial profiles smooth enough and bounded away from zero
and one,
the macroscopic density profile for our KCLG evolves under the
diffusive time
scaling according to the porous medium equation. Then we prove the
same result
for more general profiles for a slightly perturbed dynamics obtained
by adding
jumps of the Symmetric Simple Exclusion. The role of the latter is to
remove
the degeneracy of rates and at the same time they are properly slowed
down in
order not to change the macroscopic behavior. The equilibrium
fluctuations and
the magnitude of the spectral gap for this perturbed model are also
obtained.
http://arxiv.org/abs/0704.2242
---------------------------------------------------------------
5535. BROWNIAN EXCURSION AREA, WRIGHT'S CONSTANTS IN GRAPH
ENUMERATION, AND OTHER BROWNIAN AREAS
Svante Janson
This survey is a collection of various results and formulas by different
authors on the areas (integrals) of five related processes, viz.
Brownian
motion, bridge, excursion, meander and double meander; for the
Brownian motion
and bridge, which take both positive and negative values, we consider
both the
integral of the absolute value and the integral of the positive (or
negative)
part. This gives us seven related positive random variables, for
which we
study, in particular, formulas for moments and Laplace transforms; we
also give
(in many cases) series representations and asymptotics for density
functions
and distribution functions. We further study Wright's constants
arising in the
asymptotic enumeration of connected graphs; these are known to be
closely
connected to the moments of the Brownian excursion area.
The main purpose is to compare the results for these seven
Brownian areas by
stating the results in parallel forms; thus emphasizing both the
similarities
and the differences. A recurring theme is the Airy function which
appears in
slightly different ways in formulas for all seven random variables.
We further
want to give explicit relations between the many different similar
notations
and definitions that have been used by various authors. There are
also some new
results, mainly to fill in gaps left in the literature. Some short
proofs are
given, but most proofs are omitted and the reader is instead referred
to the
original sources.
http://arxiv.org/abs/0704.2289
---------------------------------------------------------------
5536. EXISTENCE OF GRAPHS WITH SUB EXPONENTIAL TRANSITIONS
PROBABILITY DECAY AND APPLICATIONS
Clement Rau (LATP)
In this paper, we present a complete proof of the construction of
graphs with
bounded valency such that the simple random walk has a return
probability at
time $n$ at the origin of order $exp(-n^{\alpha}),$ for fixed $\alpha
\in
[0,1[$ and with Folner function $exp(n^{\frac{2\alpha}{1-\alpha}})$.
We begin
by giving a more detailled proof of this result contained in (see
\cite{ershdur}). In the second part, we give an application of the
existence of
such graphs. We obtain bounds of the correct order for some
functional of the
local time of a simple random walk on an infinite cluster on the
percolation
model.
http://arxiv.org/abs/0704.2337
---------------------------------------------------------------
5537. LOCAL WELL-POSEDNESS OF MUSIELA'S SPDE WITH L\'EVY NOISE
Carlo Marinelli
We determine sufficient conditions on the volatility coefficient of
Musiela's
stochastic partial differential equation driven by an infinite
dimensional
L\'evy process so that it admits a unique local mild solution in
spaces of
functions whose first derivative is square integrable with respect to
a weight.
http://arxiv.org/abs/0704.2380
---------------------------------------------------------------
5538. MINIMIZING PROBABILITY OF RUIN AND A GAME OF STOPPING AND CONTROL
Erhan Bayraktar and Virginia R. Young
We consider three closely related problems in optimal control: (1)
minimizing
the probability of lifetime ruin when the rate of consumption is
stochastic and
when the individual can invest in a Black-Scholes financial market; (2)
minimizing the probability of lifetime ruin when the rate of
consumption is
constant but the individual can invest in two risky correlated
assets; and (3)
a controller-stopper problem: first, the controller controls the
drift and
volatility of a process in order to maximize a running reward based
on that
process; then, the stopper chooses the time to stop the running
reward and
rewards the controller a final amount at that time. We show that the
values
functions associated with these three problems are smooth and are the
unique
classical solutions of their Hamilton-Jacobi-Bellman equations. We
reveal an
interesting relationship among the value functions of the three
problems.
http://arxiv.org/abs/0704.2244
---------------------------------------------------------------
5539. STRONG SPHERICAL ASYMPTOTICS FOR ROTOR-ROUTER AGGREGATION AND
THE DIVISIBLE SANDPILE
Lionel Levine and Yuval Peres
The rotor-router model is a deterministic analogue of random walk. It
can be
used to define a deterministic growth model analogous to internal
DLA. We prove
that the asymptotic shape of this model is a Euclidean ball, in a
sense which
is stronger than our earlier work. For the shape consisting of $n=
\omega_d r^d$
sites, where $\omega_d$ is the volume of the unit ball in $\R^d$, we
show that
the inradius of the set of occupied sites is at least $r-O(\log r)$,
while the
outradius is at most $r+O(r^\alpha)$ for any $\alpha > 1-1/d$. For a
related
model, the divisible sandpile, we show that the domain of occupied
sites is a
Euclidean ball with error in the radius a constant independent of the
total
mass. For the classical abelian sandpile model in two dimensions,
with $n=\pi
r^2$ particles, we show that the inradius is at least $r/\sqrt{3}$,
and the
outradius is at most $(r+o(r))/\sqrt{2}$. This improves on bounds of
Le Borgne
and Rossin. Similar bounds apply in higher dimensions.
http://arxiv.org/abs/0704.0688
---------------------------------------------------------------
5540. ENTROPIC MEASURE AND WASSERSTEIN DIFFUSION
Max-K von Renesse and Karl-Theodor Sturm
We construct a new random probability measure on the sphere and on
the unit
interval which in both cases has a Gibbs structure with the relative
entropy
functional as Hamiltonian. It satisfies a quasi-invariance formula
with respect
to the action of smooth diffeomorphism of the sphere and the interval
respectively. The associated integration by parts formula is used to
construct
two classes of diffusion processes on probability measures (on the
sphere or
the unit interval) by Dirichlet form methods. The first one is
closely related
to Malliavin's Brownian motion on the homeomorphism group. The second
one is a
probability valued stochastic perturbation of the heat flow, whose
intrinsic
metric is the quadratic Wasserstein distance. It may be regarded as the
canonical diffusion process on the Wasserstein space.
http://arxiv.org/abs/0704.0704
---------------------------------------------------------------
5541. WEAK AND STRONG TAYLOR METHODS FOR NUMERICAL SOLUTIONS OF
STOCHASTIC DIFFERENTIAL EQUATIONS
Maria Siopacha and Josef Teichmann
We apply results of Malliavin-Thalmaier-Watanabe for strong and weak
Taylor
expansions of solutions of perturbed stochastic differential
equations (SDEs).
In particular, we work out weight expressions for the Taylor
coefficients of
the expansion. The results are applied to LIBOR market models in
order to deal
with the typical stochastic drift and with stochastic volatility. In
contrast
to other accurate methods like numerical schemes for the full SDE, we
obtain
easily tractable expressions for accurate pricing. In particular, we
present an
easily tractable alternative to ``freezing the drift'' in LIBOR
market models,
which has an accuracy similar to the full numerical scheme. Numerical
examples
underline the results.
http://arxiv.org/abs/0704.0745
---------------------------------------------------------------
5542. COMPUTATION OF POWER LOSS IN LIKELIHOOD RATIO TESTS FOR
PROBABILITY DENSITIES EXTENDED BY LEHMANN ALTERNATIVES
Lucas Gallindo and Martins Soares
We compute the loss of power in likelihood ratio tests when we test the
original parameter of a probability density extended by the first
Lehmann
alternative.
http://arxiv.org/abs/0704.0739
---------------------------------------------------------------
5543. COMPUTATION OF POWER LOSS IN LIKELIHOOD RATIO TESTS FOR
PROBABILITY DENSITIES EXTENDED BY LEHMANN ALTERNATIVES
Lucas Gallindo and Martins Soares
We compute the loss of power in likelihood ratio tests when we test the
original parameter of a probability density extended by the first
Lehmann
alternative.
http://arxiv.org/abs/0704.0739
---------------------------------------------------------------
5544. HITTING PROBABILITIES FOR SYSTEMS OF NON-LINEAR STOCHASTIC
HEAT EQUATIONS WITH MULTIPLICATIVE NOISE
Robert C. Dalang and Davar Khoshnevisan and and Eulalia Nualart
We consider a system of d non-linear stochastic heat equations in
spatial
dimension 1 driven by d-dimensional space-time white noise. The non-
linearities
appear both as additive drift terms and as multipliers of the noise.
Using
techniques of Malliavin calculus, we establish upper and lower bounds
on the
one-point density of the solution u(t,x), and upper bounds of
Gaussian-type on
the two-point density of (u(s,y),u(t,x)). In particular, this estimate
quantifies how this density degenerates as (s,y) converges to (t,x).
From these
results, we deduce upper and lower bounds on hitting probabilities of
the
process {u(t,x)}_{t \in \mathbb{R}_+, x \in [0,1]}, in terms of
respectively
Hausdorff measure and Newtonian capacity. These estimates make it
possible to
show that points are polar when d >6 and are not polar when d<6. We
also show
that the Hausdorff dimension of the range of the process is 6 when
d>6, and
give analogous results for the processes t \mapsto u(t,x) and x
\mapsto u(t,x).
Finally, we obtain the values of the Hausdorff dimensions of the
level sets of
these processes.
http://arxiv.org/abs/0704.1312
---------------------------------------------------------------
5545. LARGE PORTFOLIO LOSSES; A DYNAMIC CONTAGION MODEL
Paolo Dai Pra and Wolfgang J. Runggaldier and Elena Sartori and
Marco Tolotti
Using particle system methodologies we study the propagation of
financial
distress in a network of firms facing credit risk. We investigate the
phenomenon of a credit crisis and quantify the losses that a bank may
suffer in
a large credit portfolio. Applying a large deviation principle we
compute the
limiting distributions of the system and determine the time evolution
of the
credit quality indicators of the firms, deriving moreover the
dynamics of a
global financial health indicator. We finally describe a suitable
version of
the ``central limit theorem'' useful to study large portfolio losses.
Simulation results are provided as well as applications to portfolio
loss
distribution analysis.
http://arxiv.org/abs/0704.1348
---------------------------------------------------------------
5546. SOBOLEV SOLUTION FOR SEMILINEAR PDE WITH OBSTACLE UNDER
MONOTONICITY CONDITION
A.Matoussi and M. Xu
We prove the existence and uniqueness of the solution of a semilinear
PDE's
and also PDE's with obstacle under monotonicity condition. Moreover
we give the
probabilistic interpretation of the Sobolev's solutions in term of
Backward SDE
and reflected Backward SDE respectively.
http://arxiv.org/abs/0704.1414
---------------------------------------------------------------
5547. EXACT RETROSPECTIVE MONTE CARLO COMPUTATION OF ARITHMETIC
AVERAGE ASIAN OPTIONS
Benjamin Jourdain (CERMICS) and Mohamed Sbai (CERMICS)
Using ideas from the exact algorithm of Beskos, Papaspiliopoulos and
Roberts,
we derive an exact simulation based technique for pricing continuous
arithmetic
average Asian options in the Black and Scholes framework. Unlike
existing Monte
Carlo methods, we are no longer prone to the discretization bias
resulting from
the approximation of continuous time processes through discrete
sampling.
http://arxiv.org/abs/0704.1433
---------------------------------------------------------------
5548. LARGE DEVIATIONS OF POISSON CLUSTER PROCESSES
Charles Bordenave and Giovanni Luca Torrisi
In this paper we prove scalar and sample path large deviation
principles for
a large class of Poisson cluster processes. As a consequence, we
provide a
large deviation principle for ergodic Hawkes point processes.
http://arxiv.org/abs/0704.1463
---------------------------------------------------------------
5549. WILLIAMS' DECOMPOSITION OF THE L\'EVY CONTINUOUS RANDOM TREE
AND SIMULTANEOUS EXTINCTION PROBABILITY FOR POPULATIONS WITH NEUTRAL
MUTATIONS
Romain Abraham (MAPMO) and Jean-Fran\c{c}ois Delmas (CERMICS)
We consider an initial Eve-population and a population of neutral
mutants,
such that the total population dies out in finite time. We describe the
evolution of the Eve-population and the total population with
continuous state
branching processes, and the neutral mutation procedure can be seen
as an
immigration process with intensity proportional to the size of the
population.
First we establish a Williams' decomposition of the genealogy of the
total
population given by a continuous random tree, according to the ancestral
lineage of the last individual alive. This allows us give a closed
formula for
the probability of simultaneous extinction of the Eve-population and
the total
population.
http://arxiv.org/abs/0704.1475
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