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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.
Author(s): Philippe Briand and Fulvia Confortola
Abstract: 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
Author(s): David Nualart and Salvador Ortiz-Latorre
Abstract: 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
Author(s): Craig A. Tracy and Harold Widom
Abstract: 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
Author(s): Yana Volkovich and Nelly Litvak and Debora Donato
Abstract: 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
Author(s): L. R. G. Fontes and C. M. Newman and K. Ravishankar and E. Schertzer
Abstract: 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
Author(s): Zongxia Liang
Abstract: 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
Author(s): Scott Zrebiec
Abstract: 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
Author(s): Michael R\"ockner and Xicheng Zhang
Abstract: 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
Author(s): Xicheng Zhang
Abstract: 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
Author(s): D. Denisov and A. B. Dieker and V. Shneer
Abstract: 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
Author(s): Anne-Laure Basdevant (PMA) and Arvind Singh (PMA)
Abstract: 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
Author(s): Alexei Borodin (1) and Patrik L. Ferrari (2) and Tomohiro Sasamoto (3) ((1) Caltech, (2) WIAS Berlin, (3) Chiba University)
Abstract: 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
Author(s): Christian Baer and Frank Pfaeffle
Abstract: 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
Author(s): Nikolaos Fountoulakis
Abstract: 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
Author(s): Viorel Barbu and Giuseppe Da Prato and Michael R\"ockner
Abstract: 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
Author(s): Viorel Barbu and Giuseppe Da Prato and Michael R\"ockner
Abstract: 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
Author(s): Philippe Briand (IRMAR) and Ying Hu (IRMAR)
Abstract: 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
Author(s): M. Mania and R. Tevzadze and T. Toronjadze
Abstract: 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
Author(s): Joaquin Fontbona and Helene Guerin and Sylvie Meleard
Abstract: 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
Author(s): Daniel Gandolfo and Jean Ruiz and Daniel Ueltschi
Abstract: 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
Author(s): Vitalii A. Gasanenko
Abstract: 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
Author(s): H.J. Haubold and A.M. Mathai and R.K. Saxena
Abstract: 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
Author(s): Fabien Panloup (PMA) and Gilles Pag{\`e}s (PMA)
Abstract: 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
Author(s): Y. Git and J. W. Harris and S. C. Harris
Abstract: 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
Author(s): Anna Ja\'{s}kiewicz
Abstract: 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
Author(s): Florian Dennert and Rudolf Gr\"{u}bel
Abstract: 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
Author(s): W. Kang and R. J. Williams
Abstract: 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
Author(s): H.J. Haubold and A.M. Mathai and R.K. Saxena
Abstract: 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
Author(s): Jonathon Peterson and Ofer Zeitouni
Abstract: 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
Author(s): Ying Hu (IRMAR) and Jin Ma (Department of Mathematics) and Shige Peng (Institute of Mathematics), Song Yao (Department of Mathematics)
Abstract: 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
Author(s): Kevin Ford
Abstract: 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
Author(s): I. Bjelakovic and J.-D. Deuschel and T. Krueger and R. Seiler and Ra. Siegmund-Schultze, A. Szkola
Abstract: 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
Author(s): 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)
Abstract: 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
Author(s): Firas Rassoul-Agha and Timo Sepp\"{a}l\"{a}inen
Abstract: 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
Author(s): Albert Hanen
Abstract: 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
Author(s): Shizan Fang and Peter Imkeller and Tusheng Zhang
Abstract: 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
Author(s): Jan Bergenthum and Ludger R\"{u}schendorf
Abstract: 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
Author(s): S\'ebastien Darses (LM-Besan\c{c}on) and Ivan Nourdin (LM-Besan\c{c}on)
Abstract: 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
Author(s): Henrik Hult and Filip Lindskog
Abstract: 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
Author(s): J. du Toit and G. Peskir
Abstract: 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
Author(s): Isabelle Chalendar and Jonathan R. Partington
Abstract: 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
Author(s): Erhan Bayraktar
Abstract: 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
Author(s): Luigi Ambrosio and Giuseppe Savare and Lorenzo Zambotti
Abstract: 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
Author(s): Alain-Sol Sznitman
Abstract: 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
Author(s): Kerry M. Soileau
Abstract: 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
Author(s): A. Goldenhsluger and O. Lepski
Abstract: 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
Author(s): Wenming Hong and Ofer Zeitouni
Abstract: 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
Author(s): Jan Obloj and Martijn Pistorius
Abstract: 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
Author(s): Marie Cottrell (SAMOS and Matisse) and Tatiana Turova (DMS Lund)
Abstract: 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
Author(s): Guang-Ming Pan and Mei-Hui Guo and Wang Zhou
Abstract: 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
Author(s): George Yin and Hanqin Zhang
Abstract: 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
Author(s): Matthew O. Jones and Richard F. Serfozo
Abstract: 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
Author(s): Ralph P. Russo and Nariankadu D. Shyamalkumar
Abstract: 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
Author(s): Yongtao Guan and Stephen M. Krone
Abstract: 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
Author(s): Christian Y. Robert and Johan Segers
Abstract: 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
Author(s): Francois Baccelli and Charles Bordenave
Abstract: 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
Author(s): Franz Merkl and Silke W.W. Rolles
Abstract: 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
Author(s): Abba M. Krieger and Moshe Pollak and Ester Samuel-Cahn
Abstract: 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
Author(s): Darrell Duffie and Yeneng Sun
Abstract: 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
Author(s): Alexander I. Bufetov and Boris M. Gurevich
Abstract: 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
Author(s): Wei Wang and Jinqiao Duan
Abstract: 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
Author(s): J\'{e}r\'{e}mie Bourdon (LINA) and Damien Eveillard (LINA)
Abstract: 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
Author(s): P.Baldi and D.Marinucci and V.S.Varadarajan
Abstract: 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
Author(s): Markus Jalsenius and Kasper Pedersen
Abstract: 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
Author(s): Rados{\l}aw Adamczak and Rafa{\l} Lata{\l}a
Abstract: 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
Author(s): Miklos Bona
Abstract: 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
Author(s): Jinho Baik and Robert Buckingham and and Jeffery DiFranco
Abstract: 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
Author(s): Yuri N.Kartashov and Alexey M.Kulik
Abstract: 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
Author(s): Fulvia Confortola
Abstract: 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
Author(s): S. Bonaccorsi and F. Confortola and E. Mastrogiacomo
Abstract: 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
Author(s): Martin Keller-Ressel and Thomas Steiner
Abstract: 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
Author(s): C. Kuelske and E. Orlandi
Abstract: 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
Author(s): F. Hiai and D. Petz
Abstract: 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
Author(s): F. Hiai and D. Petz
Abstract: 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
Author(s): Nathana\"{e}l Enriquez (PMA) and Christophe Sabot (ICJ) and Olivier Zindy (PMA)
Abstract: 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
Author(s): Nathana\"{e}l Enriquez (PMA) and Christophe Sabot (ICJ) and Olivier Zindy (PMA)
Abstract: 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
Author(s): A. Gaudilliere
Abstract: 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
Author(s): Laurent Bruneau and Alain Joye and Marco Merkli
Abstract: 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
Author(s): Luigi Manca
Abstract: 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
Author(s): Norio Konno
Abstract: 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
Author(s): Vladislav Vysotsky
Abstract: 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
Author(s): Zbigniew Pucha{\l}a and Tomasz Rolski
Abstract: 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 $x_1<...t) = C h(x)t^{-\alpha}e^{-\gamma t}(1 + o(1))$ as $t\to\infty$ and
identify $C,h(x),\alpha,\gamma$ in terms of the drifts.
http://arXiv.org/abs/0704.0215.abs
Author(s): P\'eter E. Frenkel
Abstract: 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
Author(s): Francesco Caravenna and Jean-Dominique Deuschel
Abstract: 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
Author(s): Patrick Cattiaux (CMAP and LSProba) and Arnaud Guillin (LATP)
Abstract: 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
Author(s): Lung-Chi Chen and Akira Sakai
Abstract: 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
Author(s): Kasper Pedersen
Abstract: 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
Author(s): Julio Largo and Piero Tartaglia and Francesco Sciortino
Abstract: 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
Author(s): Benjamin Doerr and Tobias Friedrich
Abstract: 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
Author(s): Walter Schachermayer and Uwe Schmock and Josef Teichmann
Abstract: 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
Author(s): Bassetti Federico and Leisen Fabrizio
Abstract: 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
Author(s): F. Alberto Grunbaum
Abstract: 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
Author(s): Biao Wu
Abstract: 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
Author(s): Erhan Bayraktar and Ulrich Horst and Ronnie Sircar
Abstract: 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
Author(s): Erhan Bayraktar and Ulrich Horst and Ronnie Sircar
Abstract: 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
Author(s): J. A. D. Appleby and M. Riedle
Abstract: 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
Author(s): Erhan Bayraktar and H. Vincent Poor and Ronnie Sircar
Abstract: 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
Author(s): Erhan Bayraktar and Virginia R. Young
Abstract: 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
Author(s): Erhan Bayraktar and Masahiko Egami
Abstract: 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
Author(s): Erhan Bayraktar
Abstract: 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
Author(s): Erhan Bayraktar and Masahiko Egami
Abstract: 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
Author(s): Erhan Bayraktar and H. Vincent Poor
Abstract: 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
Author(s): Erhan Bayraktar and Masahiko Egami
Abstract: 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
Author(s): Erhan Bayraktar and Virginia R. Young
Abstract: 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
Author(s): Mika Hujo
Abstract: 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
Author(s): Revaz Tevzadze
Abstract: 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
Author(s): Kurt Johansson
Abstract: 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
Author(s): Cyril Furtlehner (INRIA Futurs) and Jean-Marc Lasgouttes (INRIA Rocquencourt), Arnaud De La Fortelle (INRIA Rocquencourt)
Abstract: 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
Author(s): Antoine Gerschenfeld and Andrea Montanari
Abstract: 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
Author(s): S Satheesh and E Sandhya
Abstract: 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
Author(s): Elise Janvresse (LMRS) and Thierry De La Rue (LMRS)
Abstract: 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
Author(s): Nabil Kahale
Abstract: 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
Author(s): Seth Sullivant
Abstract: 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
Author(s): Andrei Khrennikov
Abstract: 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
Author(s): Pascal Moyal
Abstract: 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
Author(s): Tamer Oraby
Abstract: 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
Author(s): Antal A. J\'arai and Russell Lyons
Abstract: 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
Author(s): Allan Sly
Abstract: 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
Author(s): A. Faggionato
Abstract: 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
Author(s): Christian M. Reidys
Abstract: 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
Author(s): Yaozhong Hu and David Nualart
Abstract: 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
Author(s): Dorje C. Brody and Lane P. Hughston and Andrea Macrina
Abstract: 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
Author(s): Alberto Lanconelli
Abstract: 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
Author(s): Anilesh Mohari
Abstract: 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
Author(s): Anilesh Mohari
Abstract: 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
Author(s): Peter G. Doyle and Charles M. Grinstead and J. Laurie Snell
Abstract: 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
Author(s): Constantinos Daskalakis and Alexandros G. Dimakis and Elchanan Mossel
Abstract: 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
Author(s): A. J. E. M. Janssen and J. S. H. van Leeuwaarden
Abstract: 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
Author(s): Hock Peng Chan and Tze Leung Lai
Abstract: 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
Author(s): F. Toninelli (Laboratoire de Physique and ENS Lyon and CNRS UMR 5672)
Abstract: 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
Author(s): Jean Mairesse and Fr\'{e}d\'{e}ric Math\'{e}us
Abstract: 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
Author(s): Michael Farber and Thomas Kappeler
Abstract: 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
Author(s): Aim\'e Lachal (ICJ) and Thomas Simon (DP)
Abstract: 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
Author(s): Elchanan Mossel and Allan Sly
Abstract: 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
Author(s): Juan Carlos Pardo Millan
Abstract: 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
Author(s): Mathew D. Penrose
Abstract: 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
Author(s): Jocelyne Bion-Nadal
Abstract: 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
Author(s): Ciprian Tudor (CES and SAMOS) and Soledad Torres
Abstract: 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
Author(s): Ciprian Tudor (CES and SAMOS) and Khalifa Es-Sebaiy
Abstract: 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
Author(s): Raluca Balan and Ciprian Tudor (CES and SAMOS)
Abstract: 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
Author(s): Franco Flandoli (DIPARTIMENTO Di Matematica Applicata Pisa) and Massimiliano Gubinelli (LM-Orsay), Francesco Russo (LAGA)
Abstract: 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
Author(s): Mark W. Meckes
Abstract: 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
Author(s): Feng Yu
Abstract: 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
Author(s): Boris Tsirelson
Abstract: 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
Author(s): Jason Fulman
Abstract: 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
Author(s): Terence Tao and Van Vu
Abstract: 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
Author(s): Vladislav Kargin
Abstract: 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
Author(s): S Satheesh and E Sandhya
Abstract: 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
Author(s): Dominique Bakry (LSProba) and Patrick Cattiaux (MODAL'X and CMAP) and Arnaud Guillin (LATP)
Abstract: 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
Author(s): Gil Kalai
Abstract: 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
Author(s): Anatly Vershik
Abstract: 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
Author(s): Uwe Franz and Adam Skalski
Abstract: 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
Author(s): Serban T. Belinschi and Alexandru Nica
Abstract: 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
Author(s): Iddo Ben-Ari
Abstract: 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
Author(s): Alexander Gnedin and Yuri Yakubovich
Abstract: 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
Author(s): Nicolas Fournier
Abstract: 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
Author(s): Zhen Wu and Zhiyong Yu
Abstract: 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
Author(s): Michael J. Kozdron (University of Regina)
Abstract: 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
Author(s): Bernardo Lafuerza-Guillen and Donal O'Regan and Reza Saadati
Abstract: 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
Author(s): Hiroki Sumi
Abstract: 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
Author(s): Louis-Pierre Arguin
Abstract: 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
Author(s): Shunichi Nomura and Akimichi Takemura
Abstract: 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
Author(s): Yasunori Horikoshi and Akimichi Takemura
Abstract: 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
Author(s): Xavier Bressaud and Nicolas Fournier
Abstract: 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
Author(s): 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)
Abstract: 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
Author(s): Alberto Gandolfi and Roberto Guenzani
Abstract: 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
Author(s): Benjamin Bruder (PMA) and Huyen Pham (PMA)
Abstract: 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
Author(s): M.D. Jara and C. Landim and S. Sethuraman
Abstract: 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
Author(s): David Nualart and Salvador Ortiz
Abstract: 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
Author(s): Zhiyi Chi
Abstract: 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
Author(s): Alexander Barvinok and Alex Samorodnitsky and and Alexander Yong
Abstract: 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
Author(s): Elchanan Mossel
Abstract: 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
Author(s): Thierry L\'{e}vy (DMA)
Abstract: 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
Author(s): Mikhail Lifshits and Michel Weber
Abstract: 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
Author(s): Mikhail Lifshits and Michel Weber
Abstract: 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
Author(s): Michel Weber
Abstract: 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
Author(s): Jeremie Unterberger
Abstract: 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 $ = 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
Author(s): Wei Wang and Daomin Cao and Jinqiao Duan
Abstract: 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
Author(s): Mark Rudelson and Roman Vershynin
Abstract: 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
Author(s): Jussi I. Tyhtila
Abstract: 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
Author(s): Nick Dungey
Abstract: 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
http://arXiv.org/abs/math/0703530
Author(s): Witold Bednorz
Abstract: 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
Author(s): Annalisa Cerquetti
Abstract: 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
Author(s): J.C. Pardo
Abstract: 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
Author(s): Wei Wang and Jinqiao Duan
Abstract: 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
Author(s): Margit R\"osler and Michael Voit
Abstract: 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
Author(s): Michael Bateman
Abstract: 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
Author(s): E. Fischler and Z. Schuss
Abstract: 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
Author(s): Peter McCullagh and Jim Pitman and Matthias Winkel
Abstract: 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
Author(s): Soumik Pal and Jim Pitman
Abstract: 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
Author(s): Firas Rassoul-Agha and Timo Seppalainen
Abstract: 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
Author(s): Cecile Monthus and Thomas Garel
Abstract: 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
http://arXiv.org/abs/cond-mat/0602200
Author(s): Cecile Monthus and Thomas Garel
Abstract: 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
Author(s): Cecile Monthus and Thomas Garel
Abstract: 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
Author(s): Cecile Monthus and Thomas Garel
Abstract: 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
Author(s): Cecile Monthus and Thomas Garel
Abstract: 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
$T_cT_{gap}$, 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
http://arXiv.org/abs/cond-mat/0611611
Author(s): Cecile Monthus and Thomas Garel
Abstract: 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
Author(s): Cecile Monthus and Thomas Garel
Abstract: We consider the low-temperature $T
http://arXiv.org/abs/cond-mat/0702131
Author(s): Cecile Monthus and Thomas Garel
Abstract: 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
http://arXiv.org/abs/cond-mat/0703017
Author(s): Emanuel Milman and Sasha Sodin
Abstract: 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
Author(s): Itai Benjamini and Noam Berger and Ariel Yadin
Abstract: 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
Author(s): Andreas Greven and Vlada Limic and Anita Winter
Abstract: 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
Author(s): Leonid Bogachev and Gregory Derfel and Stanislav Molchanov and and John Ockendon
Abstract: 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
Author(s): Peter G. Doyle
Abstract: 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
Author(s): Alessandro De Gregorio
Abstract: We deal with a planar random flight $\{(X(t),Y(t)),0
http://arXiv.org/abs/math/0703887
Author(s): Erhan Bayraktar and Virginia R. Young
Abstract: 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
Author(s): Itai Benjamini and Ori Gurel-Gurevich and Roey Izkovsky
Abstract: 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
Author(s): F.M.Dekking and L. van Driel
Abstract: 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
Author(s): Lorenzo Finesso and Peter Spreij
Abstract: 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
Author(s): Alexandre Belloni and Victor Chernozhukov
Abstract: 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
Author(s): Michel Fliess (LIX and Inria Futurs)
Abstract: 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
Author(s): Ad\'{a}m Tim\'{a}r
Abstract: 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
Author(s): \'{A}d\'{a}m Tim\'{a}r
Abstract: 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
Author(s): Vlad Bally
Abstract: 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
Author(s): Richard Durrett and Deena Schmidt
Abstract: 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
Author(s): Vlad Bally and Marie-Pierre Bavouzet and Marouen Messaoud
Abstract: 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
Author(s): Stephen G. Walker and Spyridon J. Hatjispyros and Theodoros Nicoleris
Abstract: 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
Author(s): Z. D. Bai and Jack W. Silverstein
Abstract: 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
Author(s): Florin Avram and Zbigniew Palmowski and Martijn R. Pistorius
Abstract: 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
Author(s): Gennadiy Averkov (University of Magdeburg) and Gabriele Bianchi (Universita` di Firenze)
Abstract: 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
Author(s): Rowan Killip
Abstract: 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
Author(s): Milton Jara and Gregorio Moreno and Alejandro F. Ramirez
Abstract: 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
Author(s): Marek Biskup and Herbert Spohn
Abstract: 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
Author(s): Amaury Lambert (FESE)
Abstract: 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
Author(s): Jean Bertoin (PMA and Dma)
Abstract: 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
Author(s): Thierry de la Rue and Roberto Fernandez and Alan D. Sokal
Abstract: 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
Author(s): Manuel Lladser and Meredith D. Betterton and Rob Knight
Abstract: 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
Author(s): Geoffrey Grimmett and Tobias Osborne and Petra Scudo
Abstract: 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
Author(s): Massimiliano Gubinelli and Jozsef Lorinczi
Abstract: 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
Author(s): Patricia Goncalves and Claudio Landim and Cristina Toninelli
Abstract: 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
Author(s): Svante Janson
Abstract: 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
Author(s): Clement Rau (LATP)
Abstract: 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
Author(s): Carlo Marinelli
Abstract: 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
Author(s): Erhan Bayraktar and Virginia R. Young
Abstract: 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
Author(s): Lionel Levine and Yuval Peres
Abstract: 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
Author(s): Max-K von Renesse and Karl-Theodor Sturm
Abstract: 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
Author(s): Maria Siopacha and Josef Teichmann
Abstract: 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
Author(s): Lucas Gallindo and Martins Soares
Abstract: 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
Author(s): Lucas Gallindo and Martins Soares
Abstract: 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
Author(s): Robert C. Dalang and Davar Khoshnevisan and and Eulalia Nualart
Abstract: 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
Author(s): Paolo Dai Pra and Wolfgang J. Runggaldier and Elena Sartori and Marco Tolotti
Abstract: 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
Author(s): A.Matoussi and M. Xu
Abstract: 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
Author(s): Benjamin Jourdain (CERMICS) and Mohamed Sbai (CERMICS)
Abstract: 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
Author(s): Charles Bordenave and Giovanni Luca Torrisi
Abstract: 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
Author(s): Romain Abraham (MAPMO) and Jean-Fran\c{c}ois Delmas (CERMICS)
Abstract: 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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