[Pas] Probability Abstracts 90
pas at www.economia.unimi.it
pas at www.economia.unimi.it
Wed Jan 4 09:55:34 CET 2006
January 4, 2006
Letter 90
Probability Abstract Service
Abstracts from Nov-1-2005 to Dec-29-2005
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3796. PROPAGATION OF FLUCTUATIONS IN BIOCHEMICAL SYSTEMS, I: LINEAR
SSC NETWORKS
David Anderson and Jonathan Mattingly and H. Frederik Nijhout and
Michael Reed
We investigate the propagation of random fluctuations through
biochemical
networks in which the concentrations of species are large enough so
that the
unperturbed problem is well-described by ordinary differential
equation. We
characterize the behavior of variance as fluctuations propagate down
chains,
study the effect of side chains and feedback loops, and investigate the
asymptotic behavior as one rate constant gets large. We also describe
how the
ideas can be applied to the study of methionine metabolism.
http://front.math.ucdavis.edu/math.PR/0510642
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3797. TRANSPORTATION TO RANDOM ZEROES BY THE GRADIENT FLOW
Fedor Nazarov and Mikhail Sodin and Alexander Volberg
We show that the basins of zeroes under the gradient flow of the random
potential U corresponding to a random Gaussian Entire Function f
partition the
complex plane into domains of equal area and that the probability
that the
diameter of a particular basin is greater than R is exponentially
small in R.
http://front.math.ucdavis.edu/math.CV/0510654
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3798. NO MULTIPLE COLLISIONS FOR MUTUALLY REPELLING BROWNIAN PARTICLES
Emmanuel C\'{e}pa (MAPMO) and Dominique L\'{e}pingle (MAPMO)
Brownian particles in electrostatic interaction may pairwise collide
when the
interaction parameter is small. But multiple collisions are never
possible.
http://front.math.ucdavis.edu/math.PR/0511445
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3799. A PERMUTATION TEST FOR MATCHING AND ITS ASYMPTOTIC DISTRIBUTION
Larry Goldstein and Yosef Rinott
We consider a permutation method for testing whether observations
given in
their natural pairing exhibit an unusual level of similarity in
situations
where any two observations may be similar at some unknown baseline
level. Under
a null hypotheses where there is no distinguished pairing of the
observations,
a normal approximation with explicit bounds and rates is presented for
determining approximate critical test levels.
http://front.math.ucdavis.edu/math.ST/0511427
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3800. COMBINATORICS AND DISTRIBUTIONS OF PARTIAL INJECTIONS
Olexandr Ganyushkin and Volodymyr Mazorchuk
We obtain several combinatorial results about chains, cycles and
orbits of
the elements of the symmetric inverse semigroup $\IS_n$ and the set
$T_n$ of
nilpotent elements in $\IS_n$. We also get some estimates for the
growth of
$|\IS_n|$ and $|T_n|$, and study random products of elements from $
\IS_n$.
http://front.math.ucdavis.edu/math.CO/0511431
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3801. MULTIPLE ORTHOGONAL POLYNOMIALS OF MIXED TYPE AND NON-
INTERSECTING BROWNIAN MOTIONS
E. Daems and A.B.J. Kuijlaars
We present a generalization of multiple orthogonal polynomials of
type I and
type II, which we call multiple orthogonal polynomials of mixed type.
Some
basic properties are formulated, and a Riemann-Hilbert problem for
the multiple
orthogonal polynomials of mixed type is given. We derive a
Christoffel-Darboux
formula for these polynomials using the solution of the Riemann-Hilbert
problem. The main motivation for studying these polynomials comes
from a model
of non-intersecting one-dimensional Brownian motions with a given
number of
starting points and endpoints. The correlation kernel for the
positions of the
Brownian paths at any intermediate time coincides with the
Christoffel-Darboux
kernel for the multiple orthogonal polynomials of mixed type with
respect to
Gaussian weights.
http://front.math.ucdavis.edu/math.CA/0511470
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3802. AN ORIENTED COMPETITION MODEL ON Z_{+}^2
George Kordzakhia and Steven Lalley
We consider a two-type oriented competition model on the first
quadrant of
the two-dimensional integer lattice. Each vertex of the space may
contain only
one particle of either Red type or Blue type. A vertex flips to the
color of a
randomly chosen southwest nearest neighbor at exponential rate 2. At
time zero
there is one Red particle located at (1,0) and one Blue particle
located at
(0,1). The main result is a partial shape theorem: Denote by R(t) and
B(t) the
red and blue regions at time t. Then (i) eventually the upper half of
the unit
square contains no points of B(t)=t, and the lower half no points of R
(t)=t;
and (ii) with positive probability there are angular sectors rooted
at (1,1)
that are eventually either red or blue. The second result is
contingent on the
uniform curvature of the boundary of the corresponding Richardson shape.
http://front.math.ucdavis.edu/math.PR/0511504
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3803. BERRY ESSEEN BOUNDS FOR COMBINATORIAL CENTRAL LIMIT THEOREMS
AND PATTERN OCCURRENCES, USING ZERO AND SIZE BIASING
Larry Goldstein
Berry Esseen type bounds to the normal, based on zero- and size-bias
couplings, are derived using Stein's method. The zero biasing bounds are
illustrated with an application to combinatorial central limit
theorems where
the random permutation has either the uniform distribution or one
which is
constant over permutations with the same cycle type and having no
fixed points.
The size biasing bounds are applied to the occurrences of fixed
relatively
ordered sub-sequences (such as rising sequences) in a random
permutation, and
to the occurrences of patterns, extreme values, and subgraphs on
finite graphs.
http://front.math.ucdavis.edu/math.PR/0511510
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3804. RANDOM DENSE COUNTABLE SETS: CHARACTERIZATION BY INDEPENDENCE
Boris Tsirelson
A random dense countable set is characterized (in distribution) by
independence and stationarity. Two examples are `Brownian local
minima' and
`unordered infinite sample'. They are identically distributed; the
former ad
hoc proof of this fact is now superseded by a general result.
http://front.math.ucdavis.edu/math.PR/0511011
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3805. STOCHASTIC INTEGRAL WITH RESPECT TO CYLINDRICAL WIENER PROCESS
Anna Karczewska
This paper is devoted to a construction of the stochastic It\^o
integral with
respect to infinite dimensional cylindrical Wiener process. The
construction
given is an alternative one to that introduced by DaPrato and Zabczyk
[3]. The
connection of the introduced integral with the integral defined by
Walsh [9] is
provided as well.
http://front.math.ucdavis.edu/math.PR/0511512
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3806. RANDOM TREES AND APPLICATIONS
Jean-Francois Le Gall
We discuss several connections between discrete and continuous random
trees.
In the discrete setting, we focus on Galton-Watson trees under various
conditionings. In particular, we present a simple approach to Aldous'
theorem
giving the convergence in distribution of the contour process of
conditioned
Galton-Watson trees towards the normalized Brownian excursion. We
also briefly
discuss applications to combinatorial trees. In the continuous
setting, we use
the formalism of real trees, which yields an elegant formulation of the
convergence of rescaled discrete trees towards continuous objects. We
explain
the coding of real trees by functions, which is a continuous version
of the
well-known coding of discrete trees by Dyck paths. We pay special
attention to
random real trees coded by Brownian excursions, and in a particular
we provide
a simple derivation of the marginal distributions of the CRT. The
last section
is an introduction to the theory of the Brownian snake, which
combines the
genealogical structure of random real trees with independent spatial
motions.
We introduce exit measures for the Brownian snake and we present some
applications to a class of semilinear partial differential equations.
http://front.math.ucdavis.edu/math.PR/0511515
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3807. EXPONENTIAL FUNCTIONALS OF BROWNIAN MOTION, I: PROBABILITY LAWS
AT FIXED TIME
Hiroyuki Matsumoto Marc Yor
This paper is the first part of our survey on various results about the
distribution of exponential type Brownian functionals defined as an
integral
over time of geometric Brownian motion. Several related topics are also
mentioned.
http://front.math.ucdavis.edu/math.PR/0511517
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3808. EXPONENTIAL FUNCTIONALS OF BROWNIAN MOTION, II: SOME RELATED
DIFFUSION PROCESSES
Hiroyuki Matsumoto Marc Yor
This is the second part of our survey on exponential functionals of
Brownian
motion. We focus on the applications of the results about the
distributions of
the exponential functionals, which have been discussed in the first
part.
Pricing formula for call options for the Asian options, explicit
expressions
for the heat kernels on hyperbolic spaces, diffusion processes in random
environments and extensions of L\'evy's and Pitman's theorems are
discussed.
http://front.math.ucdavis.edu/math.PR/0511519
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3809. A VARIATION EMBEDDING THEOREM AND APPLICATIONS
Peter Friz and Nicolas Victoir
Fractional Sobolev spaces, also known as Besov or Slobodetzki spaces,
arise
in many areas of analysis, stochastic analysis in particular. We
prove an
embedding into certain q-variation spaces and discuss a few
applications. First
we show q-variation regularity of Cameron-Martin paths associated to
fractional
Brownian motion and other Volterra processes. This is useful, for
instance, to
establish large deviations for enhanced fractional Brownian motion.
Second, the
q-variation embedding, combined with results of rough path theory,
provides a
different route to a regularity result for stochastic differential
equations by
Kusuoka. Third, the embedding theorem works in a non-commutative
setting and
can be used to establish Hoelder/variation regularity of rough paths.
http://front.math.ucdavis.edu/math.PR/0511520
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3810. GIANT COMPONENTS IN BIASED GRAPH PROCESSES
Gideon Amir and Ori Gurel-Gurevich and Eyal Lubetzky and Amit Singer
A random graph process, $\Gorg[1](n)$, is a sequence of graphs on $n$
vertices which begins with the edgeless graph, and where at each step
a single
edge is added according to a uniform distribution on the missing
edges. It is
well known that in such a process a giant component (of linear size)
typically
emerges after $(1+o(1))\frac{n}{2}$ edges (a phenomenon known as
``the double
jump''), i.e., at time $t=1$ when using a timescale of $n/2$ edges in
each
step.
We consider a generalization of this process, $\Gorg[K](n)$, which
gives a
weight of size 1 to missing edges between pairs of isolated vertices,
and a
weight of size $K \in [0,\infty)$ otherwise. This corresponds to a
case where
links are added between $n$ initially isolated settlements, where the
probability of a new link in each step is biased according to whether
or not
its two endpoint settlements are still isolated.
Combining methods of \cite{SpencerWormald} with analytical
techniques, we
describe the typical emerging time of a giant component in this process,
$t_c(K)$, as the singularity point of a solution to a set of
differential
equations. We proceed to analyze these differential equations and obtain
properties of $\Gorg$, and in particular, we show that $t_c(K)$ strictly
decreases from 3/2 to 0 as $K$ increases from 0 to $\infty$, and that
$t_c(K) =
\frac{4}{\sqrt{3K}}(1 + o(1))$. Numerical approximations of the
differential
equations agree both with computer simulations of the process $\Gorg
(n)$ and
with the analytical results.
http://front.math.ucdavis.edu/math.PR/0511526
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3811. FOURIER TRANSFORM OF A GAUSSIAN MEASURE ON THE HEISENBERG GROUP
Matyas Barczy and Gyula Pap
An explicit formula is derived for the Fourier transform of a Gaussian
measure on the Heisenberg group at the Schrodinger representation.
Using this
explicit formula, necessary and sufficient conditions are given for the
convolution of two Gaussian measures to be a Gaussian measure.
http://front.math.ucdavis.edu/math.PR/0511016
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3812. THE SPATIAL $\LAMBDA$-COALESCENT
Vlada Limic and Anja Sturm
This paper extends the notion of the $\la$-coalescent of Pitman
(1999) to the
spatial setting. The partition elements of the spatial $\Lambda$-
coalescent
migrate in a (finite) geographical space and may only coalesce if
located at
the same site of the space. We characterize the $\Lambda$-coalescents
that come
down from infinity, in an analogous way to Schweinsberg (2000).
Surprisingly,
all spatial coalescents that come down from infinity, also come down
from
infinity in a uniform way. This enables us to study space-time
asymptotics of
spatial $\Lambda$-coalescents on large tori in $d\ge 3$ dimensions.
Our results
generalize and strengthen those of Greven et al. (2005), who studied the
spatial Kingman coalescent in this context.
http://front.math.ucdavis.edu/math.PR/0511536
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3813. THE REALIZATION OF POSITIVE RANDOM VARIABLES VIA ABSOLUTELY
CONTINUOUS TRANSFORMATIONS OF MEASURE ON WIENER SPACE
D. Feyel and A.S. Ustunel and M. Zakai
Let \mu be a Gaussian measure on some measurable space {W = {w},
\calB (W)}
and let \nu be a measure on the same space which is absolutely
continuous with
respect to \nu. The paper surveys results on the problem of
constructing a
transformation T on the W space such that Tw = w+u(w) where u takes
values in
the Cameron-Martin space and the image of \mu under T is \mu. In
addition we
ask for the existence of transformations T belonging to some particular
classes.
http://front.math.ucdavis.edu/math.PR/0511545
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3814. RANDOM WALK MODELS AND PROBABILISTIC TECHNIQUES FOR
INHOMOGENEOUS POLYMER CHAINS
Francesco Caravenna
Modeling of polymer chains has received a lot of attention in
mathematics. In
fact, probabilistic models that naturally arise in statistical
mechanics have
been widely studied by mathematicians for the very challenging and novel
problems that they pose. The physical situation that we consider in
this thesis
is that of a polymer in the proximity of an interface between two
selective
solvents, in the case when the interaction of the monomers with the
solvents
and the interface may vary from monomer to monomer (inhomogeneous
polymer). In
interesting cases thee is a phase transition between a state in which
the
polymer sticks very close to the interface (localized regime) and a
state in
which it wanders away from it (delocalized regime). The mechanism
underlying
such a transition is an energy/entropy competition.
Our task has been to study random walk models of polymer chains
with the
purpose of understanding this competition in a deep and quantitative
way.
Despite the fact that the definition of these models is extremely
elementary,
their analysis is not simple at all, and several interesting
questions are
still open. In this Ph.D. thesis we present new results that answer
some of
these questions. The analysis performed has required the application
of a wide
range of techniques, including large deviations, concentration
inequalities,
renewal theory, fluctuation theory for random walks. A numerical and
statistical study has been performed too. Finally we prove a local limit
theorem for random walks conditioned to stay positive.
http://front.math.ucdavis.edu/math.PR/0511561
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3815. ON CONSTRAINED ANNEALED BOUNDS FOR PINNING AND WETTING MODELS
Francesco Caravenna and Giambattista Giacomin
The free energy of quenched disordered systems is bounded above by
the free
energy of the corresponding annealed system. This bound may be
improved by
applying the annealing procedure, which is just Jensen inequality,
after having
modified the Hamiltonian in a way that the quenched expressions are left
unchanged. This procedure is often viewed as a partial annealing or as a
constrained annealing, in the sense that the term that is added may be
interpreted as a Lagrange multiplier on the disorder variables.
In this note we point out that, for a family of models, some of
which have
attracted much attention, the multipliers of the form of empirical
averages of
local functions cannot improve on the basic annealed bound from the
viewpoint
of characterizing the phase diagram. This class of multipliers is the
one that
is suitable for computations and it is often believed that in this
class one
can approximate arbitrarily well the quenched free energy.
http://front.math.ucdavis.edu/math.PR/0511562
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3816. A MODIFIED VERSION OF FROZEN PERCOLATION ON THE BINARY TREE
R.Brouwer
We consider the following, intuitively described process: at time
zero, all
sites of a binary tree are at rest. Each site becomes activated at a
random
uniform [0,1] time, independent of the other sites. As soon as a site
is in an
infinite cluster of activated sites, this cluster of activated sites
freezes.
The main question is whether a process like this exists. Aldous
[Ald00] proved
that this is the case for a slightly different version of frozen
percolation.
In this paper we construct a process that fits the intuitive
description and
discuss some properties.
http://front.math.ucdavis.edu/math.PR/0511021
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3817. DIRECTED PERCOLATION IN TWO DIMENSIONS: AN EXACT SOLUTION
L. C. Chen and F. Y. Wu
We consider a directed percolation process on an ${\cal M}$ x ${\cal N}$
rectangular lattice whose vertical edges are directed upward with an
occupation
probability y and horizontal edges directed toward the right with
occupation
probabilities x and 1 in alternate rows. We deduce a closed-form
expression for
the percolation probability P(x,y), the probability that one or more
directed
paths connect the lower-left and upper-right corner sites of the
lattice. It is
shown that P(x,y) is critical in the aspect ratio $a = {\cal M}/{\cal
N}$ at a
value $a_c =[1-y^2-x(1-y)^2]/2y^2$ where P(x,y) is discontinuous, and
the
critical exponent of the correlation length for $a < a_c$ is $\nu=2$.
http://front.math.ucdavis.edu/cond-mat/0511296
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3818. ON THE LIMITING DISTRIBUTION FOR THE LONGEST ALTERNATING
SEQUENCE IN A RANDOM PERMUTATION
Harold Widom
Recently Richard Stanley initiated a study of the distribution of the
length
as(w) of the longest alternating subsequence in a random permutation
w from the
symmetric group $S_n$. Among other things he found an explicit
formula for the
generating function (on n and k) for the probability that as(w) is at
most k
and conjectured that the distribution, suitably centered and
normalized, tended
to a Gaussian with variance 8/45. In this note we present a proof of the
conjecture based on the generating function.
http://front.math.ucdavis.edu/math.CO/0511533
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3819. LINEAR FUNCTIONS ON THE CLASSICAL MATRIX GROUPS
Elizabeth Meckes
Let $M$ be a random matrix in the orthogonal group $\O_n$, distributed
according to Haar measure, and let $A$ be a fixed $n\times n$ matrix
over $\R$
such that $\tr(AA^t)=n$. Then the total variation distance of the random
variable $\tr(AM)$ to standard normal is bounded by $2\sqrt{3}/(n-1)
$, and this
rate is sharp up to the constant. Analogous results are obtained for
$M$ a
random unitary matrix and $A$ a fixed $n\times n$ matrix over $\C$.
The proofs
are via an improvement of Stein's method of exchangeable pairs which
makes use
of the continuous nature of the symmetries of the classical matrix
groups.
http://front.math.ucdavis.edu/math.PR/0509441
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3820. ZERO BIASING AND A DISCRETE CENTRAL LIMIT THEOREM
Larry Goldstein and Aihua Xia
We introduce a new family of distributions to approximate $\prob(W\in
A)$ for
$A\subset\{...,-2,-1,0,1,2,...\}$ and $W$ a sum of independent
integer-valued
random variables $\xi_1$, $\xi_2$, $...$, $\xi_n$ with finite second
moments,
where with large probability $W$ is not concentrated on a lattice of
span
greater than 1. The well-known Berry--Esseen theorem states that for
$Z$ a
normal random variable with mean $\mean(W)$ and variance $\var(W)$, $
\prob(Z
\in A)$ provides a good approximation to $\prob(W \in A)$ for $A$ of
the form
$(-\infty,x]$. However, for more general $A$ such as the set of all even
numbers, the normal approximation becomes unsatisfactory and it is
desirable to
have an appropriate discrete, non-normal, distribution which
approximates $W$
in total variation, and a discrete version of the Berry--Esseen
theorem to
bound the error. In this paper, using the concept of zero biasing for
discrete
random variables [cf Goldstein and Reinert (2005)], we introduce a
new family
of discrete distributions and provide a discrete version of the
Berry--Esseen
theorem showing how members of the family approximate the
distribution of a sum
$W$ of integer valued variables in total variation.
http://front.math.ucdavis.edu/math.PR/0509444
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3821. ON A CLASS OF STOCHASTIC SEMILINEAR PDE'S
Luigi Manca
We consider stochastic semilinear partial differential equations with
Lipschitz nonlinear terms. We prove existence and uniqueness of an
invariant
measure and the existence of a solution for the corresponding Kolmogorov
equation in the space $L^2(H;\nu)$, where $\nu$ is the invariant
measure. We
also prove the closability of the derivative operator and an
integration by
parts formula. Finally, under boundness conditions on the nonlinear
term, we
prove a Poincar\'e inequality, a logarithmic Sobolev inequality and the
ipercontractivity of the transition semigroup.
http://front.math.ucdavis.edu/math.PR/0509446
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3822. A CENTRAL LIMIT THEOREM AND HIGHER ORDER RESULTS FOR THE
ANGULAR BISPECTRUM
D. Marinucci
The angular bispectrum of spherical random fields has recently gained an
enormous importance, especially in connection with statistical
inference on
cosmological data. In this paper, we provide expressions for its
moments of
arbitrary order and we use these results to establish a multivariate
central
limit theorem and higher order approximations. The results rely upon
combinatorial methods from graph theory and a detailed investigation
for the
asymptotic behaviour of Clebsch-Gordan coefficients; the latter are
widely used
in representation theory and quantum theory of angular momentum.
http://front.math.ucdavis.edu/math.PR/0509430
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3823. FLUCTUATIONS OF THE FRONT IN A STOCHASTIC COMBUSTION MODEL
Francis Comets and Jeremy Quastel and Alejandro F. Ramirez
We consider an interacting particle system on the one dimensional
lattice
$\bf Z$ modeling combustion. The process depends on two integer
parameters
$2\le a<M<\infty$. Particles move independently as continuous time
simple
symmetric random walks except that 1. When a particle jumps to a site
which has
not been previously visited by any particle, it branches into $a$
particles; 2.
When a particle jumps to a site with $M$ particles, it is
annihilated. We start
from a configuration where all sites to the left of the origin have been
previously visited and study the law of large numbers and central
limit theorem
for $r_t$, the rightmost visited site at time $t$.
The proofs are based on the construction of a renewal structure
leading to a
definition of regeneration times for which good tail estimates can be
performed.
http://front.math.ucdavis.edu/math.PR/0511025
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3824. THE CONFORMALLY INVARIANT MEASURE ON SELF-AVOIDING LOOPS
Wendelin Werner
We show that there exists (up to multiplicative constants) a unique and
natural measure on simple loops on Riemann surfaces, such that the
measure is
conformally invariant and also invariant under restriction (i.e. the
measure on
a Riemann surface S' that is contained in another Riemann surface S,
is just
the measure on S restricted to those loops that stay in S'). We then
study some
of its properties and consequences concerning outer boundaries of
critical
percolation clusters and Brownian loops.
http://front.math.ucdavis.edu/math.PR/0511605
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3825. THRESHOLD FOR MONOTONE SYMMETRIC PROPERTIES THROUGH A
LOGARITHMIC SOBOLEV INEQUALITY
Rapha\"el Rossignol
Threshold phenomena are investigated under a general approach, following
Talagrand, Friedgut and Kalai. The general upper bound for the
threshold width
of symmetric monotone properties is improved. This follows from a new
lower
bound on the maximal influence of a variable on a Boolean function.
The method
of proof is based upon a well known logarithmic Sobolev inequality on
the
discrete cube. This new bound is shown to be asymptotically optimal.
http://front.math.ucdavis.edu/math.PR/0511607
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3826. ENUMERATING CONTINGENCY TABLES VIA RANDOM PERMANENTS
Alexander Barvinok
Given m positive integers R=(r_i), n positive integers C=(c_j) such
that sum
r_i = sum c_j =N, and mn non-negative weights W=(w_ij), we consider
the total
weight T(R, C; W) of non-negative integer matrices (contingency tables)
D=(d_ij) with the row sums r_i, column sums c_j, and the weight of D
equal to
the product w_ij^{d_ij}$. We present a randomized algorithm of a
polynomial in
N complexity which approximates T(R,C; W) within a factor of (2 pi N)^
{-1/2} (2
pi t)^{N/2t} e^{N/12t^2} where t=max{min r_i, min c_j}. In many
cases, this
approximation provides an asymptotically accurate estimate of ln T(R,
C; W).
The idea of the algorithm is to express T(R,C; W) as the expectation
of the
permanent of an NxN random matrix with exponentially distributed
entries and
approximate the expectation by the integral of an efficiently computable
log-concave function on R^{mn}.
http://front.math.ucdavis.edu/math.CO/0511596
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3827. THE PROBABILITY OF A RUN
Mark B. Villarino
We prove the explicit formula for the probability of a run of r
successes in
n trials.
http://front.math.ucdavis.edu/math.PR/0511652
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3828. A SIMPLE THEORY FOR THE STUDY OF SDES DRIVEN BY A FRACTIONAL
BROWNIAN MOTION, IN DIMENSION ONE
Ivan Nourdin (LPMA)
In this paper, we will focus - in dimension one - on the SDEs of the
type
dX\_t=s(X\_t)dB\_t+b(X\_t)dt where B is a fractional Brownian motion.
Our
principal motivation is to describe one of the simplest theory - from
our point
of view - allowing to study this SDE, and this for any Hurst index H
between 0
and 1. We will consider several definitions of solution and we will
study, for
each one of them, in which condition one has existence and
uniqueness. Finally,
we will examine the convergence or not of the canonical scheme
associated to
our SDE, when the integral with respect to fBm is defined using the
Russo-Vallois symmetric integral.
http://front.math.ucdavis.edu/math.PR/0511027
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3829. OPTIMAL FLOW THROUGH THE DISORDERED LATTICE
David J. Aldous (U.C. Berkeley)
Consider routing traffic on the $N \times N$ torus, simultaneously
between
all source-destination pairs, to minimize the cost $\sum_e c(e)f^2(e)
$, where
$f(e)$ is the volume of flow across edge $e$ and the $c(e)$ form an
i.i.d.
random environment. We prove existence of a rescaled $N \to \infty$
limit
constant for minimum cost, by comparison with an appropriate
analogous problem
about minimum-cost flows across a $M \times M$ subsquare of the lattice.
http://front.math.ucdavis.edu/math.PR/0511694
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3830. FRAGMENTATION ASSOCIATED TO LEVY PROCESSES USING SNAKE
Romain Abraham (MAPMO) and Jean-Francois Delmas (CERMICS)
We consider the height process of a Levy process with no negative
jumps, and
its associated continuous tree representation. Using Levy snake tools
developed
by Duquesne and Le Gall, with an underlying Poisson process, we
construct a
fragmentation process, which in the stable case corresponds to the
self-similar
fragmentation described by Miermont. For the general fragmentation
process we
compute a family of dislocation measures as well as the law of the
size of a
tagged fragment. We also give a special Markov property for the snake
which is
interesting in itself.
http://front.math.ucdavis.edu/math.PR/0511702
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3831. KOLMOGOROV EQUATIONS IN INFINITE DIMENSIONS: WELL-POSEDNESS
AND REGULARITY OF SOLUTIONS, WITH APPLICATIONS TO STOCHASTIC
GENERALIZED BURGERS
EQUATIONS
Michael R\"ockner and Zeev Sobol
We develop a new method to uniquely solve a large class of heat
equations, so
called Kolmogorov equations in infinitely many variables. The
equations are
analyzed in spaces of sequentially weakly continuous functions
weighted by
proper (Lyapunov type) functions. This way for the first time the
solutions are
constructed everywhere without exceptional sets for equations with
possibly
non-locally Lipschitz drifts. Apart from general analytic interest,
the main
motivation is to apply this to uniquely solve martingale problems in
the sense
of Stroock-Varadhan given by stochastic partial differential
equations from
hydrodynamics, such as the stochastic Navier-Stokes equations. In
this paper
this is done in the case of the stochastic generalized Burgers equation.
Uniqueness is shown in the sense of Markov flows.
http://front.math.ucdavis.edu/math.PR/0511708
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3832. A LONG RANGE DEPENDENCE STABLE PROCESS AND AN INFINITE
VARIANCE BRANCHING SYSTEM
Tomasz Bojdecki and Luis G. Gorostiza and Anna Talarczyk
We prove a functional limit theorem for the rescaled occupation time
fluctuations of a (d,\alpha,\beta)-branching particle system
(particles moving
in R^d according to a symmetric \alpha-stable Levy process, branching
law in
the domain of attraction of a (1+\beta)-stable law, 0<\beta<1,
uniform Poisson
initial state) in the case of intermediate dimensions, \alpha/\beta <
d <
\alpha(1+\beta)/\beta. The limit is a process of the form K\lambda
\xi, where K
is a constant, \lambda is the Lebesgue measure on R^d, and \xi =
(\xi_t)_{t\geq
0} is a (1+\beta)-stable process which has long range dependence.
There are two
long range dependence regimes, one for all \beta>d/(d+\alpha), which
coincides
with the case of finite variance branching (\beta=1), and another one
for
\beta\leq d/(d+\alpha), where the long range dependence depends on
the value of
\beta. The long range dependence is characterized by a dependence
exponent
\kappa which describes the asymptotic behavior of the codifference of
increments of \xi on intervals far apart, and which is d/\alpha for
the first
case and (1+\beta-d/(d+\alpha))d/\alpha for the second one. The
convergence
proofs use techniques of S'(R^d)-valued processes.
http://front.math.ucdavis.edu/math.PR/0511739
---------------------------------------------------------------
3833. THE PROCESS OF MOST RECENT COMMON ANCESTORS IN AN EVOLVING
COALESCENT
P. Pfaffelhuber and A. Wakolbinger
In a population of constant size, whose family sizes evolve as Wright-
Fisher
diffusions, all individuals alive at time $t$ have a most recent common
ancestor (MRCA) who lived at time $A(t)$, say. The process $(A(t))$ has
piecewise constant paths. At each jump time $E_n$, a new MRCA takes
over, who
lived at time $B_n:=A(E_n)$. We construct the random sequence $(B_n,
E_n)$ in
terms of a look-down process and investigate its dynamics as well as
that of
$(A(t))$. In particular, we find the joint distribution of the
waiting time
from $t$ to the next MRCA change and of the time when this next MRCA
will have
lived.
http://front.math.ucdavis.edu/math.PR/0511743
---------------------------------------------------------------
3834. THE FULL BROWNIAN WEB AS SCALING LIMIT OF STOCHASTIC FLOWS
Luiz Renato Fontes Charles M. Newman
In this paper we construct an object which we call the full Brownian web
(FBW) and prove that the collection of all space-time trajectories of
a class
of one-dimensional stochastic flows converges weakly, under diffusive
rescaling, to the FBW. The (forward) paths of the FBW include the
coalescing
Brownian motions of the ordinary Brownian web along with bifurcating
paths.
Convergence of rescaled stochastic flows to the FBW follows from general
characterization and convergence theorems that we present here
combined with
earlier results of Piterbarg.
http://front.math.ucdavis.edu/math.PR/0511029
---------------------------------------------------------------
3835. OCCUPATION TIME FLUCTUATIONS OF AN INFINITE VARIANCE BRANCHING
SYSTEM IN LARGE DIMENSIONS
Tomasz Bojdecki and Luis G. Gorostiza and Anna Talarczyk
We prove limit theorems for rescaled occupation time fluctuations of a
(d,alpha,beta)-branching particle system (particles moving in R^d
according to
a spherically symmetric alpha-stable Levy process, (1+beta)-branching,
0<beta<1, uniform Poisson initial state), in the cases of critical
dimension,
d=alpha(1+beta)/beta, and large dimensions, d>alpha(1+beta)/beta. The
fluctuation processes are continuous but their limits are stable
processes with
independent increments, which have jumps. The convergence is in the
sense of
finite-dimensional distributions, and also of space-time random fields
(tightness does not hold in the usual Skorohod topology). The results
are in
sharp contrast with those for intermediate dimensions, alpha/beta < d <
d(1+beta)/beta, where the limit process is continuous and has long range
dependence (this case is studied by Bojdecki et al, 2005). The limit
process is
measure-valued for the critical dimension, and S'(R^d)-valued for large
dimensions. We also raise some questions of interpretation of the
different
types of dimension-dependent results obtained in the present and
previous
papers in terms of properties of the particle system.
http://front.math.ucdavis.edu/math.PR/0511745
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3836. ASYMPTOTIC BEHAVIOR OF EDGE-REINFORCED RANDOM WALKS
Franz Merkl and Silke Rolles
In this article, we study linearly edge-reinforced random walk on
general
multi-level ladders for large initial edge weights. For infinite
ladders, we
show that the process can be represented as a random walk in a random
environment, given by random weights on the edges. The edge weights
decay
exponentially in space. The process converges to a stationary
process. We
provide asymptotic bounds for the range of the random walker up to a
given
time, showing that it localizes much more than an ordinary random
walker. The
random environment is described in terms of an infinite-volume Gibbs
measure.
http://front.math.ucdavis.edu/math.PR/0511750
---------------------------------------------------------------
3837. QUANTITATIVE CONCENTRATION INEQUALITIES ON SAMPLE PATH SPACE
FOR MEAN FIELD INTERACTION
Fran\c{c}ois Bolley (UMPA-ENSL)
We consider a system of particles experiencing diffusion and mean field
interaction, and study its behaviour when the number of particles
goes to
infinity. We derive non-asymptotic large deviation bounds measuring the
concentration of the empirical measure of the paths of the particles
around its
limit. The method is based on a coupling argument, strong integrability
estimates on the paths in Holder norm, and some general concentration
result
for the empirical measure of identically distributed independent paths.
http://front.math.ucdavis.edu/math.PR/0511752
---------------------------------------------------------------
3838. ROSENTHAL TYPE INEQUALITIES FOR FREE CHAOS
Marius Junge and Javier Parcet and Quanhua Xu
Let $\mathcal{A}$ denote the reduced amalgamated free product of a
family
$\mathsf{A}_1, \mathsf{A}_2, ..., \mathsf{A}_n$ of von Neumann
algebras over a
von Neumann subalgebra $\Be$ with respect to normal faithful conditional
expectations $\Es_k: \mathsf{A}_k \to \Be$. We investigate the norm in
$L_p(\Al)$ of homogeneous polynomials of a given degree $d$. We first
generalize Voiculescu's inequality to arbitrary degree $d \ge 1$ and
indices $1
\le p \le \infty$. This can be regarded as a free analogue of the
classical
Rosenthal inequality. Our second result is a length-reduction formula
from
which we generalize recent results of Pisier, Ricard and the authors.
All
constants in our estimates are independent of $n$ so that we may
consider
infinitely many free factors. As applications, we study square
functions of
free martingales. More precisely we show that, in contrast with the
Khintchine
and Rosenthal inequalities, the free analogue of the Burkholder-Gundy
inequalities does not hold on $L_\infty(\Al)$. At the end of the
paper we also
consider Khintchine type inequalities for Shlyakhtenko's generalized
circular
systems.
http://front.math.ucdavis.edu/math.OA/0511732
---------------------------------------------------------------
3839. SPATIAL AND NON-SPATIAL STOCHASTIC MODELS FOR IMMUNE RESPONSE
Rinaldo Schinazi and Jason Schweinsberg
We study some simple mathematical models designed to test the following
hypothesis: can a pathogen escape the immune system only because of
its high
probability of mutation? We propose both spatial and non-spatial
models. In all
of our models, we assume that pathogens can mutate, leading to the
appearance
of new types of pathogens. We also assume that the immune system is
able to get
rid of all the pathogens of a given type at once but that it
recognizes only
one type at a time.
http://front.math.ucdavis.edu/math.PR/0512009
---------------------------------------------------------------
3840. COLOURING POWERS OF CYCLES FROM RANDOM LISTS
Michael Krivelevich and Asaf Nachmias
Let $C_n^k$ be the $k$-th power of a cycle on $n$ vertices (i.e. the
vertices
of $C_n^k$ are those of the $n$-cycle, and two vertices are connected
by an
edge if their distance along the cycle is at most $k$). For each
vertex draw
uniformly at random a subset of size $c$ from a base set $S$ of size
$s=s(n)$.
In this paper we solve the problem of determining the asymptotic
probability of
the existence of a proper colouring from the lists for all fixed
values of
$c,k$, and growing $n$.
http://front.math.ucdavis.edu/math.CO/0512004
---------------------------------------------------------------
3841. COLOURING COMPLETE BIPARTITE GRAPHS FROM RANDOM LISTS
Michael Krivelevich and Asaf Nachmias
Let $K_{n,n}$ be the complete bipartite graph with $n$ vertices in
each side.
For each vertex draw uniformly at random a list of size $k$ from a
base set $S$
of size $s=s(n)$. In this paper we estimate the asymptotic
probability of the
existence of a proper colouring from the random lists for all fixed
values of
$k$ and growing $n$. We show that this property exhibits a sharp
threshold for
$k\geq 2$ and the location of the threshold is precisely $s(n)=2n$
for $k=2$,
and approximately $s(n)=\frac{n}{2^{k-1}\ln 2}$ for $k\geq 3$.
http://front.math.ucdavis.edu/math.CO/0512010
---------------------------------------------------------------
3842. INCREASING AND DECREASING SUBSEQUENCES OF PERMUTATIONS AND
THEIR VARIANTS
Richard P. Stanley
We survey the theory of increasing and decreasing subsequences of
permutations. Enumeration problems in this area are closely related
to the RSK
algorithm. The asymptotic behavior of the expected value of the
length is(w) of
the longest increasing subsequence of a permutation w of 1,2,...,n
was obtained
by Vershik-Kerov and (almost) by Logan-Shepp. The entire limiting
distribution
of is(w) was then determined by Baik, Deift, and Johansson. These
techniques
can be applied to other classes of permutations, such as involutions,
and are
related to the distribution of eigenvalues of elements of the
classical groups.
A number of generalizations and variations of increasing/decreasing
subsequences are discussed, including the theory of pattern
avoidance, unimodal
and alternating subsequences, and crossings and nestings of matchings
and set
partitions.
http://front.math.ucdavis.edu/math.CO/0512035
---------------------------------------------------------------
3843. LIMIT VELOCITY AND ZERO-ONE LAWS FOR DIFFUSIONS IN RANDOM
ENVIRONMENT
Laurent Goergen
This article is accepted for publication in the "Annals of Applied
Probability". We prove that multi-dimensional diffusions in random
environment
have a limiting velocity which takes at most two different values.
Further, in
the two-dimensional case we show that for any direction, the
probability to
escape to infinity in this direction equals either zero or one.
Combined with
our results on the limiting velocity, this implies a strong law of large
numbers in two dimensions.
http://front.math.ucdavis.edu/math.PR/0512061
---------------------------------------------------------------
3844. A MICROSCOPIC INTERPRETATION FOR ADAPTIVE DYNAMICS TRAIT
SUBSTITUTION SEQUENCE MODELS
Nicolas Champagnat (WIAS)
We consider an interacting particle Markov process for Darwinian
evolution in
an asexual population with non-constant population size, involving a
linear
birth rate, a density-dependent logistic death rate, and a
probability $\mu$ of
mutation at each birth event. We introduce a renormalization
parameter $K$
scaling the size of the population, which leads, when $K\to+\infty$,
to a
deterministic dynamics for the density of individuals holding a given
trait. By
combining in a non-standard way the limits of large population ($K\to+
\infty$)
and of small mutations ($\mu\to 0$), we prove that a time scales
separation
between the birth and death events and the mutation events occurs and
that the
interacting particle microscopic process converges for finite
dimensional
distributions to the biological model of evolution known as the
``monomorphic
trait substitution sequence'' model of adaptive dynamics, which
describes the
Darwinian evolution in an asexual population as a Markov jump process
in the
trait space.
http://front.math.ucdavis.edu/math.PR/0512063
---------------------------------------------------------------
3845. FUNCTIONAL INEQUALITIES FOR PARTICLE SYSTEMS ON POLISH SPACES
Michael R\"ockner and Feng-Yu Wang
Various Poincare-Sobolev type inequalities are studied for a
reaction-diffusion model of particle systems on Polish spaces. The
systems we
consider consist of finite particles which are killed or produced at
certain
rates, while particles in the system move on the Polish space
interacting with
one another (i.e. diffusion). Thus, the corresponding Dirichlet form,
which we
call reaction-diffusion Dirichlet form, consists of two parts: the
diffusion
part induced by certain Markov processes on the product spaces $E^n
(n \geq 1)$
which determine the motion of particles, and the reaction part
induced by a
$Q$-process on $\mathbb Z_+$ and a sequence of reference probability
measures,
where the $Q$-process determines the variation of the number of
particles and
the reference measures describe the locations of newly produced
particles. We
prove that the validity of Poincare and weak Poincare inequalities are
essentially due to the pure reaction part, i.e. either of these
inequalities
holds if and only if it holds for the pure reaction Dirichlet form, or
equivalently, for the corresponding $Q$-process. But under a mild
condition,
stronger inequalities rely on both parts: the reaction-diffusion
Dirichlet form
satisfies a super Poincare inequality (e.g. the log-Sobolev
inequality) if and
only if so do both the corresponding $Q$-process and the diffusion part.
Explicit estimates of constants in the inequalities are derived.
Finally, some
specific examples are presented to illustrate the main results.
http://front.math.ucdavis.edu/math.PR/0512100
---------------------------------------------------------------
3846. JOINT ASYMPTOTIC BEHAVIOR OF LOCAL AND OCCUPATION TIMES
Endre Cs\'{a}ki and Ant\'{o}nia F\"{o}ldes and P\'al R\'ev\'esz
Considering a simple symmetric random walk in dimension $d\geq 3$, we
study
the almost sure joint asymptotic behavior of two objects: first the
local times
of a pair of neighboring points, then the local time of a point and the
occupation time of the surface of the unit ball around it.
http://front.math.ucdavis.edu/math.PR/0511049
---------------------------------------------------------------
3847. INFINITELY DIVISIBLE DISTRIBUTIONS FOR RECTANGULAR FREE
CONVOLUTION: CLASSIFICATION AND MATRICIAL INTERPRETATION
Florent Benaych-Georges (DMA)
In a previous paper (called "Rectangular random matrices. Related
covolution"), we defined, for $\lambda \in [0,1]$, the rectangular free
convolution with ratio $\lambda$. Here, we investigate the related
notion of
infinite divisiblity, which happens to be closely related the classical
infinite divisibility: there exists a bijection between the set of
classical
symmetric infinitely divisible distributions and the set of infinitely
divisible distributions with respect to this convolution, which
preserves limit
theorems. We give an interpretation of this correspondance in term of
random
matrices: we construct distributions on sets of complex rectangular
matrices
which give rise to random matrices with singular laws (i.e. uniform
distributions on their singular values) going from the symmetric
classical
infinitely divisible distributions to their images by the previously
mentioned
bijection when the dimensions go from one to infinity in a ratio $
\lambda$.
http://front.math.ucdavis.edu/math.OA/0512080
---------------------------------------------------------------
3848. RECTANGULAR RANDOM MATRICES, RELATED FREE ENTROPY AND FREE
FISHER'S INFORMATION
Florent Benaych-Georges (DMA)
We prove that independent rectangular random matrices, when embedded
in a
space of larger square matrices, are asymptotically free with
amalgamation over
a commutative finite dimensional subalgebra $D$ (under an hypothesis
of unitary
invariance). Then we consider elements of a finite von Neumann algebra
containing $D$, which have kernel and range projection in $D$. We
associate
them a free entropy with the microstates approach, and a free Fisher's
information with the conjugate variables approach. Both give rise to
optimization problems whose solutions involve freeness with
amalgamation over
$D$. It could be a first proposition for the study of operators between
different Hilbert spaces with the tools of free probability. As an
application,
we prove a result of freeness with amalgamation between the two parts
of the
polar decomposition of $R$-diagonal elements with non trivial kernel.
http://front.math.ucdavis.edu/math.OA/0512081
---------------------------------------------------------------
3849. OPTIMAL CONTROL OF A LARGE DAM
Vyacheslav M. Abramov
A large dam model is an object of study of this paper. The parameters
$L^{lower}$ and $L^{upper}$ are its lower and upper levels,
$L=L^{upper}-L^{lower}$ is large, and if a current level of water is
between
these bounds, then the dam is assumed to be in normal state. Passage
one or
other bound leads to damage. It is assumed that input stream of water is
described by a Poisson process, while the output stream is state-
dependent (the
exact formulation of the problem is given in the paper). Let $L_t$
denote the
dam level at time $t$, and let $p_1=\lim_{t\to\infty}\mathbf{P}\{L_t=
L^{lower}\}$, $p_2=\lim_{t\to\infty}\mathbf{P}\{L_t> L^{upper}\}$
exist. Then
the expected long-run damage $J=p_1J_1+p_2J_2$ for the long time
interval $T$
proportional to $L$ ($J_1$ and $J_2$ are the corresponding damage
costs per
time $T$ associated with passage the bounds) is a performance
measure, and the
aim of the paper is to choose the parameter of output stream (exactly
specified
in the paper) minimizing $J$.
http://front.math.ucdavis.edu/math.PR/0512118
---------------------------------------------------------------
3850. QUASI-PRODUCT FORMS FOR LEVY-DRIVEN FLUID NETWORKS
K. Debicki and A. B. Dieker and T. Rolski
We study stochastic tree fluid networks driven by a multidimensional
Levy
process. We are interested in (the joint distribution of) the steady-
state
content in each of the buffers, the busy periods, and the idle
periods. To
investigate these fluid networks, we relate the above three
quantities to
fluctuations of the input Levy process by solving a multidimensional
Skorokhod
problem. This leads to the analysis of the distribution of the
componentwise
maximums, the corresponding epochs at which they are attained, and the
beginning of the first last-passage excursion. Using the notion of
splitting
times, we are able to find their Laplace transforms. It turns out
that, if the
components of the Levy process are `ordered', the Laplace transform
has a
so-called quasi-product form.
The theory is illustrated by working out special cases, such as
tandem
networks and priority queues.
http://front.math.ucdavis.edu/math.PR/0512119
---------------------------------------------------------------
3851. ASYMPYOTIC EXPANSIONS FOR INFINITE WEIGHTED CONVOLUTIONS OF
LIGHT SUBEXPONENTIAL DISTRIBUTIONS
Ph. Barbe (CNRS) and W.P. McCormick (UGA)
We establish some asymptotic expansions for infinite weighted
convolutions of
distributions having light subexponential tails. Examples are
presented, some
showing that in order to obtain an expansion with two significant
terms, one
needs to have a general way to calculate higher order expansions, due to
possible cancellations of terms. An algebraic methodology is employed
to obtain
the results.
http://front.math.ucdavis.edu/math.PR/0512141
---------------------------------------------------------------
3852. BACKWARD STOCHATIC DIFFERENTIAL EQUATIONS II
Fabrice Blache (LMA-Clermont)
In a preceding article, we have studied a generalization of the
problem of
finding a martingale on a manifold whose terminal value is known.
This article
completes the results obtained in the first article by providing
uniqueness and
existence theorems in a general framework (in particular if positive
curvatures
are allowed), still using differential geometry tools.
http://front.math.ucdavis.edu/math.PR/0512145
---------------------------------------------------------------
3853. DISTRIBUTION OF EIGENVALUES FOR THE ENSEMBLE OF REAL SYMMETRIC
PALINDROMIC TOEPLITZ MATRICES
Adam Massey and Steven J. Miller and John Sinsheimer
Consider the ensemble of real symmetric Toeplitz matrices, each
independent
entry an i.i.d. random variable chosen from a fixed probability
distribution p
of mean 0, variance 1, and finite higher moments. Previous
investigations
showed that the limiting spectral measure (the density of normalized
eigenvalues) converges (weakly and almost surely), independent of p,
to a
distribution which is almost the Gaussian. The deviations from Gaussian
behavior can be interpreted as arising from obstructions to solutions of
Diophantine equations. We show that these obstructions vanish if
instead one
considers real symmetric palindromic Toeplitz matrices (matrices
where the
first row is a palindrome), and the resulting spectral measures converge
(weakly and almost surely) to the Gaussian.
http://front.math.ucdavis.edu/math.PR/0512146
---------------------------------------------------------------
3854. ASYMPTOTIC PROPERTIES OF POWER VARIATIONS OF L\'{E}VY PROCESSES
Jean Jacod (IMJ)
We determine the asymptotic behavior of the realized power
variations, or
more generally of sums of a given test function evaluated at the
successive
increments of a L\'{e}vy process. One can completely elucidate the
first order
behavior (convergence in probability, possibly after normalization).
As for the
associated CLT, one can show some versions of it, but only in a
limited number
of cases. In some other cases, a CLT just does not exist.
http://front.math.ucdavis.edu/math.PR/0511052
---------------------------------------------------------------
3855. THE FAIR AND RANDOM MAXIMAL DIVISION OF "PIZZA"
Floyd E. Brown and Anant P. Godbole
Consider n straight line cuts of a circular pizza made so as to
maximize the
number of pieces. We investigate how fair such a maximal division may
be and
how many slices are obtained if the cuts are successfully made with a
certain
probability.
http://front.math.ucdavis.edu/math.PR/0512177
---------------------------------------------------------------
3856. MULTI-SCALING OF THE $N$-POINT DENSITY FUNCTION FOR COALESCING
BROWNIAN MOTIONS
R. Munasinghe and R. Rajesh and R. Tribe and O. Zaboronski
This paper gives a derivation for the large time asymptotics of the $n
$-point
density function of a system of coalescing Brownian motions on $\bf{R}$.
http://front.math.ucdavis.edu/math.PR/0512179
---------------------------------------------------------------
3857. STATISTICAL MECHANICAL SYSTEMS ON COMPLETE GRAPHS, INFINITE
EXCHANGEABILITY, FINITE EXTENSIONS AND A DISCRETE FINITE MOMENT PROBLEM
Thomas Liggett and Jeffrey Steif and Balint Toth
We show that a large collection of statistical mechanical systems with
quadratically represented Hamiltonians on the complete graph can be
extended to
infinite exchangeable processes. This includes all ferromagnetic
Ising, Potts
and Heisenberg models. By de Finetti's theorem, this is equivalent to
showing
that these probability measures can be expressed as averages of product
measures. We provide examples showing that ``ferromagnetism'' is not
however in
itself sufficient and also study in some detail the Ising model with an
additional 3-body interaction. Finally, we study the question of how
much the
antiferromagnetic Ising model can be extended. In this direction, we
obtain
sharp asymptotic results via a solution to a new moment problem. We
also obtain
a ``formula'' for the extension which is valid in many cases.
http://front.math.ucdavis.edu/math.PR/0512191
---------------------------------------------------------------
3858. FELLER PROPERTY AND INFINITESIMAL GENERATOR OF THE EXPLORATION
PROCESS
Romain Abraham (MAPMO) and Jean-Francois Delmas (CERMICS)
We consider the exploration process associated to the continuous
random tree
(CRT) built using a Levy process with no negative jumps. This process
has been
studied by Duquesne, Le Gall and Le Jan. This measure-valued Markov
process is
a useful tool to study CRT as well as super-Brownian motion with general
branching mechanism. In this paper we prove this process is Feller,
and we
compute its infinitesimal generator on exponential functionals and
give the
corresponding martingale.
http://front.math.ucdavis.edu/math.PR/0512195
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3859. STRICTLY STABLE DISTRIBUTIONS ON CONVEX CONES
Youri Davydov and Ilya Molchanov and Sergei Zuyev
Using the LePage representation, a strictly stable random element in
a Banach
space with $\alpha\in(0,2)$ can be represented as a sum of points of
a Poisson
process. This point process is union-stable, i.e. the union of its two
independent copies coincides in distribution with the rescaled
original point
process. These concepts makes sense in any convex cone, i.e. in a
commutative
semigroup equipped with multiplication by numbers, and lead to a
construction
of stable laws in general cones by means of the LePage series. The
corresponding limit theorem shows that random samples (or binomial point
processes) converge in distribution to the union-stable Poisson point
process,
and so yields a limit theorem for normalised sums of random elements
with
$\alpha$-stable limit for $\alpha\in(0,1)$.
By using the technique of harmonic analysis on semigroups we
characterise
distributions of $\alpha$-stable random elements and show how
possible values
of $\alpha$ relate to the properties of the semigroup and the
corresponding
scaling operation, in particular, their distributivity properties.
The approach
developed in the paper not only makes it possible to handle stable
distributions in rather general cones (like spaces of sets or
measures), but
also provides an alternative way to prove classical limit theorems
and deduce
the LePage representation for strictly stable random vectors in
Banach spaces.
http://front.math.ucdavis.edu/math.PR/0512196
---------------------------------------------------------------
3860. ON TIME INHOMOGENEOUS CONTROLLED DIFFUSION PROCESSES IN DOMAINS
Hongjie Dong and N.V. Krylov
Time inhomogeneous controlled diffusion processes in both cylindrical
and
non-cylindrical domains are considered. Bellman's principle and its
applications to proving the continuity of value functions are
investigated.
http://front.math.ucdavis.edu/math.PR/0512200
---------------------------------------------------------------
3861. THE CRITICAL RANDOM GRAPH, WITH MARTINGALES
Asaf Nachmias and Yuval Peres
We give a short proof that the largest component of the random graph
$G(n,
1/n)$ is of size approximately $n^{2/3}$. The proof gives explicit
bounds for
the probability that the ratio is very large or very small.
http://front.math.ucdavis.edu/math.PR/0512201
---------------------------------------------------------------
3862. BALLS-IN-BINS WITH FEEDBACK AND BROWNIAN MOTION
Roberto Oliveira
In a balls-in-bins process with feedback, balls are sequentially
thrown into
bins so that the probability that a bin with n balls obtains the next
ball is
proportional to f(n) for some function f. A commonly studied case
where there
are two bins and f(n) = n^p for p > 0, and our goal is to study the fine
behavior of this process with two bins and a large initial number t
of balls.
Perhaps surprisingly, Brownian Motions are an essential part of both our
proofs.
For p>1/2, it was known that with probability 1 one of the bins
will lead the
process at all large enough times. We show that if the first bin
starts with
t+\lambda\sqrt{t} balls (for constant \lambda\in \R), the probability
that it
always or eventually leads has a non-trivial limit depending on \lambda.
For p\leq 1/2, it was known that with probability 1 the bins will
alternate
in leadership. We show, however, that if the initial fraction of
balls in one
of the bins is >1/2, the time until it is overtaken by the remaining
bin scales
like \Theta({t^{1+1/(1-2p)}}) for p<1/2 and \exp(\Theta{t}) for
p=1/2. In fact,
the overtaking time has a non-trivial distribution around the scaling
factors,
which we determine explicitly.
Our proofs use a continuous-time embedding of the balls-in-bins
process (due
to Rubin) and a non-standard approximation of the process by Brownian
Motion.
The techniques presented also extend to more general functions f.
http://front.math.ucdavis.edu/math.PR/0510648
---------------------------------------------------------------
3863. ALMOST SURE ASYMPTOTICS FOR A DIFFUSION PROCESS IN A DRIFTED
BROWNIAN POTENTIAL
Alexis Devulder (PMA)
We study a one-dimensional diffusion process in a drifted Brownian
potential.
We characterize the upper functions of its hitting times in the sense
of Paul
L\'evy, and determine the lower limits in terms of an iterated
logarithm law.
http://front.math.ucdavis.edu/math.PR/0511053
---------------------------------------------------------------
3864. LARGE DEVIATION PRINCIPLE FOR ENHANCED GAUSSIAN PROCESSES
Peter Friz and Nicolas Victoir
We study large deviation principles for Gaussian processes lifted to
the free
nilpotent group of step N. We apply this to a large class of Gaussian
processes
lifted to geometric rough paths. A large deviation principle for
enhanced
(fractional) Brownian motion, in Hoelder- or modulus topology,
appears as
special case.
http://front.math.ucdavis.edu/math.PR/0512213
---------------------------------------------------------------
3865. FELLER PROCESSES ON NON-LOCALLY COMPACT SPACES
Tomasz Szarek
We introduce the ergodic condition which assures the existence of an
invariant measure for Feller processes defined on an arbitrary
complete and
separable metric space.
http://front.math.ucdavis.edu/math.PR/0512221
---------------------------------------------------------------
3866. TAIL BEHAVIOUR OF MULTIPLE RANDOM INTEGRALS AND U-STATISTICS
Peter Major
This paper contains sharp estimates about the distribution of
multiple random
integrals of functions of several variables with respect to a normalized
empirical measure, about the distribution of U-statistics and multiple
Wiener-Ito integrals with respect to a white noise. It also contains
good
estimates about the supremum of appropriate classes of such integrals or
U-statistics. The proof of most results is omitted, I have
concentrated on the
explanation of their content and the picture behind them. I also
tried to
explain the reason for the investigation of such questions. My goal
was to
yield such a presentation of the results which a non-expert also can
understand, and not only on a formal level.
http://front.math.ucdavis.edu/math.PR/0512238
---------------------------------------------------------------
3867. CRITICAL SCALING FOR THE SIMPLE SIS STOCHASTIC EPIDEMIC
R. G. Dolgoarshinnykh Steven P. Lalley
We exhibit a scaling law for the critical SIS stochastic epidemic: If
at time
0 the population consists of square root N infected and N - square
root N
susceptible individuals, then when time and number currently infected
are both
scaled by square root N, the resulting process converges, for large
N, to a
diffusion process related to the Feller diffusion by a change of
drift. As a
consequence, the rescaled size of the epidemic has a limit law that
coincides
with that of a first-passage time for the standard Ornstein-
Uhlenbeck process.
These results are the analogues for the SIS epidemic of results of
Martin-Lof
for the simple SIR epidemic.
http://front.math.ucdavis.edu/math.PR/0512252
---------------------------------------------------------------
3868. STRONG SOLUTIONS OF STOCHASTIC GENERALIZED POROUS MEDIA
EQUATIONS: EXISTENCE, UNIQUENESS AND ERGODICITY
Giuseppe Da Prato and Boris L. Rozovskii and Michael R\"ockner and
Feng-Yu Wang
Explicit conditions are presented for the existence, uniqueness and
ergodicity of the strong solution to a class of generalized
stochastic porous
media equations. Our estimate of the convergence rate is sharp
according to the
known optimal decay for the solution of the classical (deterministic)
porous
medium equation.
http://front.math.ucdavis.edu/math.PR/0512259
---------------------------------------------------------------
3869. HARMONIC CONTINUOUS TIME BRANCHING MOMENTS
Didier Piau
We show that the mean inverse populations of nondecreasing, square
integrable, continuous time branching processes decrease to zero like
the
inverse of their mean population if and only if the initial
population k is
greater than a threshold m, which is at least one. If furthermore k
is greater
than a second threshold m', which is at least m, the normalized mean
inverse
population is at most 1/(k-m'). We express m and m' as explicit
functionals of
the reproducing distribution, we discuss some analogues for discrete
time
branching processes, and we link these results to the behavior of random
products involving i.i.d. nonnegative sums.
http://front.math.ucdavis.edu/math.PR/0511058
---------------------------------------------------------------
3870. GLOBAL REGULARITY AND BOUNDS FOR SOLUTIONS OF PARABOLIC
EQUATIONS FOR PROBABILITY MEASURES
Vladimir I. Bogachev and Michael R\"ockner and Stanislav V.
Shaposhnikov
Given a second order parabolic operator
$$
Lu(t,x)
:=\frac{\partial u(t,x)}{\partial t}
+ a^{ij}(t,x)\partial_{x_i}\partial_{x_j}u(t,x)
+ b^i(t,x)\partial_{x_i}u(t,x),
$$ we consider the weak parabolic equation $L^{*}\mu=0$ for Borel
probability
measures on $(0,1)\times\mathbb{R}^d$. The equation is understood as the
equality
$$
\int_{(0,1)\times\mathbb{R}^d} Lu d\mu =0
$$ for all smooth functions $u$ with compact support
in~$(0,1)\times\mathbb{R}^d$. This equation is satisfied for the
transition
probabilities of the diffusion process associated with~$L$.
We show that under broad assumptions $\mu$ has the form $\mu=
\varrho(t,x) dt
dx$, where the function $x\mapsto \varrho(t,x)$ is Sobolev, $|\nabla_x
\varrho(x,t)|^2/\varrho(t,x)$ is Lebesgue integrable over
$[0,\tau]\times\mathbb{R}^d$, and $\varrho\in L^p([0,\tau]\times
\mathbb{R}^d)$
for all $p\in [1,+\infty)$ and $\tau<1$. Moreover, a sufficient
condition for
the uniform boundedness of $\varrho$ on $[0,\tau]\times\mathbb{R}^d$
is given.
http://front.math.ucdavis.edu/math.PR/0512264
---------------------------------------------------------------
3871. CHAOTIC STATES AND STOCHASTIC INTEGRATION IN QUANTUM SYSTEMS
V. P. Belavkin
Quantum chaotic states over a noncommutative monoid, a unitalization
of a
noncommutative Ito algebra parametrizing a quantum stochastic Levy
process, are
described in terms of their infinitely divisible generating
functionals over
the monoid-valued processes on an atomless `space-time' set. A canonical
decomposition of the logarithmic conditionally posive-definite
generating
functional is constructed in a pseudo-Euclidean space, given by a
quadruple
defining the monoid triangular operator representation and a cyclic zero
pseudo-norm state in this space. It is shown that the exponential
representation in the corresponding pseudo-Fock space yields the
infinitely-divisible generating functional with respect to the
exponential
state vector, and its compression to the Fock space defines the cyclic
infinitly-divisible representation associated with the Fock vacuum
state. The
structure of states on an arbitrary Ito algebra is studied with two
canonical
examples of quantum Wiener and Poisson states. A generalized quantum
stochastic
nonadapted multiple integral is explicitly defined in Fock scale, its
continuity and quantum stochastic differentiability is proved. A unified
non-adapted and functional quantum Ito formula is discovered and
established
both in weak and strong sense, and the multiplication formula on the
exponential Ito algebra is found for the relatively bounded kernel-
operators in
Fock scale. The unitarity and projectivity properties of nonadapted
quantum
stochastic linear differential equations are studied, and their
solution is
constructed for the locally bounded nonadapted generators in terms of
the
chronological products in the underlying kernel algebra canonically
represented
by triangular operators in the pseudo-Fock space.
http://front.math.ucdavis.edu/math.PR/0512265
---------------------------------------------------------------
3872. WEAK SOLUTIONS TO THE STOCHASTIC POROUS MEDIA EQUATION VIA
KOLMOGOROV
Viorel Barbu and Vladimir I. Bogachev and Giuseppe Da Prato and
Michael R\"ockner
A stochastic version of the porous medium equation with coloured
noise is
studied. The corresponding Kolmogorov equation is solved in the space
$L^2(H,\nu)$ where $\nu$ is an infinitesimally excessive measure.
Then a weak
solution is constructed.
http://front.math.ucdavis.edu/math.PR/0512266
---------------------------------------------------------------
3873. EXPLICIT FORMULAS FOR THE MOMENTS OF THE SOJOURN TIME IN THE M/
G/1 PROCESSOR SHARING QUEUE WITH PERMANENT JOBS
S.F.Yashkov
We give some representation about recent achievements in analysis of the
M/G/1 queue with egalitarian processor sharing discipline (EPS). The new
formmulas are derived for the j-th moments (j=1,2,...) of the
(conditional)
stationary sojourn time in the M/G/1--EPS queue with K (K=0,1,2,...)
permanent
jobs of infinite size. We discuss also how to simplify the
computations of the
moments.
http://front.math.ucdavis.edu/math.PR/0512281
---------------------------------------------------------------
3874. A PREDICTIVE THEORY OF GAMES
David H. Wolpert
Conventional noncooperative game theory hypothesizes that the joint
strategy
of a set of players in a game must satisfy an "equilibrium concept".
All other
joint strategies are considered impossible; the only issue is what
equilibrium
concept is "correct". This hypothesis violates the desiderata underlying
probability theory. Indeed, probability theory renders moot the
problem of what
equilibrium concept is correct - every joint strategy can arise with
non-zero
probability. Rather than a first-principles derivation of an equilibrium
concept, game theory requires a first-principles derivation of a
distribution
over joint (mixed) strategies. This paper shows how information
theory can
provide such a distribution over joint strategies. If a scientist
external to
the game wants to distill such a distribution to a point prediction,
that
prediction should be set by decision theory, using their (!) loss
function. So
the predicted joint strategy - the "equilibrium concept" - varies
with the
external scientist's loss function. It is shown here that in many
games, having
a probability distribution with support restricted to Nash equilibria
- as
stipulated by conventional game theory - is impossible. It is also
show how to:
i) Derive an information-theoretic quantification of a player's
degree of
rationality; ii) Derive bounded rationality as a cost of computation;
iii)
Elaborate the close formal relationship between game theory and
statistical
physics; iv) Use this relationship to extend game theory to allow
stochastically varying numbers of players.
http://front.math.ucdavis.edu/nlin.AO/0512015
---------------------------------------------------------------
3875. INFINITE DIMENSIONAL ITO ALGEBRAS OF QUANTUM WHITE NOISE
V. P. Belavkin
A simple axiomatic characterization of the general (infinite
dimensional,
noncommutative) Ito algebra is given and a pseudo-Euclidean fundamental
representation for such algebra is described. The notion of Ito B*-
algebra,
generalizing the C*-algebra is defined to include the Banach infinite
dimensional Ito algebras of quantum Brownian and quantum Levy motion,
and the
B*-algebras of vacuum and thermal quantum noise are characterized. It
is proved
that every Ito algebra is canonically decomposed into the orthogonal
sum of
quantum Brownian (Wiener) algebra and quantum Levy (Poisson) algebra. In
particular, every quantum thermal noise is the orthogonal sum of a
quantum
Wiener noise and a quantum Poisson noise as it is stated by the Levy-
Khinchin
theorem in the classical case.
http://front.math.ucdavis.edu/math.PR/0512288
---------------------------------------------------------------
3876. POSITIVE DEFINITE GERMS OF QUANTUM STOCHASTIC PROCESSES
V. P. Belavkin
A new notion of stochastic germs for quantum processes is introduced
and a
characterisation of the stochastic differentials for positive
definite (PD)
processes is found in terms of their germs for arbitrary Ito algebra. A
representation theorem, giving the pseudo-Hilbert dilation for the
germ-matrix
of the differential, is proved. This suggests the general form of
quantum
stochastic evolution equations with respect to the Poisson (jumps),
Wiener
(diffusion) or general quantum noise.
http://front.math.ucdavis.edu/math.PR/0512289
---------------------------------------------------------------
3877. ON STOCHASTIC GENERATORS OF POSITIVE DEFINITE EXPONENTS
V. P. Belavkin
A characterisation of quantum stochastic positive definite (PD)
exponent is
given in terms of the conditional positive definiteness (CPD) of their
form-generator. The pseudo-Hilbert dilation of the stochastic form-
generator
and the pre-Hilbert dilation of the corresponding dissipator is
found. The
structure of quasi-Poisson stochastic generators giving rise to a
quantum
stochastic birth processes is studied.
http://front.math.ucdavis.edu/math.PR/0512290
---------------------------------------------------------------
3878. POISSON KERNEL AND GREEN FUNCTION OF THE BALL IN REAL
HYPERBOLIC SPACES
T. Byczkowski and J. Malecki
Let $(X_t)_{t\geq0}$ be the $n$-dimensional hyperbolic Brownian
motion, that
is the diffusion on the real hyperbolic space $\D^n$ having the
Laplace-Beltrami operator as its generator. The aim of the paper is
to derive
the formulas for the Gegenbauer transform of the Poisson kernel and
the Green
function of the ball for the process $(X_t)_{t\geq0}$. Under some
additional
hypotheses we give the formulas for the Poisson kernel itself. In
particular,
we provide formulas in $\D^4$ and $\D^6$ spaces for the Poisson
kernel and the
Green function as well.
http://front.math.ucdavis.edu/math.PR/0512294
---------------------------------------------------------------
3879. RANDOM HOMEOMORPHISMS AND FOURIER EXPANSIONS - THE POINTWISE
BEHAVIOR
Gady Kozma
Let phi be a Dubins-Freedman random homeomorphism on [0,1] derived
from the
base measure uniform on the vertical line x=1/2, and let f be a periodic
function satisfying that
|f(x)-f(0)| = o(1/log log log 1/x).
Then the Fourier expansion of f composed with phi converges at 0 with
probability 1. In the condition on f, o cannot be replaced by O.
Also we deduce some 0-1 laws for this kind of problems.
http://front.math.ucdavis.edu/math.CA/0511036
---------------------------------------------------------------
3880. BINOMIAL UPPER BOUNDS ON GENERALIZED MOMENTS AND TAIL
PROBABILITIES OF (SUPER)MARTINGALES WITH DIFFERENCES BOUNDED FROM ABOVE
Iosif Pinelis
Let (S_0,S_1,...) be a supermartingale relative to a nondecreasing
sequence
of sigma-algebras H_0,H_1,..., with S_0\le0 almost surely (a.s.) and
differences X_i:=S_i-S_{i-1}. Suppose that X_i\le d and Var(X_i|H_
{i-1})\le
\si_i^2 a.s. for every i=1,2,..., where d>0 and \si_i>0 are non-random
constants. Let T_n:=Z_1+...+Z_n, where Z_1,...,Z_n are i.i.d. r.v.'s
each
taking on only two values, one of which is d, and satisfying the
conditions
E(Z_i)=0 and Var(Z_i)=\si^2:=(\si_1^2+...+\si_n^2)/n. Then, based on a
comparison inequality between generalized moments of S_n and T_n for
a rich
class of generalized moment functions, the tail comparison inequality
P(S_n \ge
y) \le c P^{\lin,\lc}(T_n \ge y+h/2)\quad\forall y\in\R is obtained,
where
c:=e^2/2=3.694..., h:=d+\si^2/d, and the function y\mapsto P^{\lin,
\lc}(T_n >
y) is the least log-concave majorant of the linear interpolation of
the tail
function y\mapsto P(T_n \ge y) over the lattice of all points of the
form nd+kh
(k\in\Z). An explicit formula for P^{\lin,\lc}(T_n\ge y+h/2) is
given. Another,
similar bound is given under somewhat different conditions. It is
shown that
these bounds improve significantly upon known bounds.
http://front.math.ucdavis.edu/math.PR/0512301
---------------------------------------------------------------
3881. LOCAL STRUCTURE OF RANDOM QUADRANGULATIONS
Maxim Krikun (IEC)
This paper is an adaptation of a method used in math.PR/0311127 to
the model
of random quadrangulations. We prove local weak convergence of
uniform measures
on quadrangulations and show that local growth of quadrangulation is
governed
by certain critical time-reversed branching process. As an
intermediate result
we calculate a biparametric generating function for certain class of
quadrangulations with boundary.
http://front.math.ucdavis.edu/math.PR/0512304
---------------------------------------------------------------
3882. LARGE SYSTEMS OF PATH-REPELLENT BROWNIAN MOTIONS IN A TRAP AT
POSITIVE TEMPERATURE
Stefan Adams and Jean-Bernard Bru and Wolfgang Koenig
We study a model of $ N $ mutually repellent Brownian motions under
confinement to stay in some bounded region of space. Our model is
defined in
terms of a transformed path measure under a trap Hamiltonian, which
prevents
the motions from escaping to infinity, and a pair-interaction
Hamiltonian,
which imposes a repellency of the $N$ paths. In fact, this
interaction is an
$N$-dependent regularisation of the Brownian intersection local
times, an
object which is of independent interest in the theory of stochastic
processes.
The time horizon (interpreted as the inverse temperature) is kept
fixed. We
analyse the model for diverging number of Brownian motions in terms
of a large
deviation principle. The resulting variational formula is the
positive-temperature analogue of the well-known Gross-Pitaevskii
formula, which
approximates the ground state of a certain dilute large quantum
system; the
kinetic energy term of that formula is replaced by a probabilistic
energy
functional.
This study is a continuation of the analysis in \cite{ABK04} where we
considered the limit of diverging time (i.e., the zero-temperature
limit) with
fixed number of Brownian motions, followed by the limit for diverging
number of
motions.
\bibitem[ABK04]{ABK04} {\sc S.~Adams, J.-B.~Bru} and {\sc W.~K
\"onig},
\newblock Large deviations for trapped interacting Brownian particles
and
paths, \newblock {\it Ann. Probab.}, to appear (2004).
http://front.math.ucdavis.edu/math.PR/0512305
---------------------------------------------------------------
3883. A FUNCTIONAL CENTRAL LIMIT THEOREM FOR A CLASS OF URN MODELS
Gopal K Basak and Amites Dasgupta
We construct an independent increments Gaussian process associated to
a class
of multicolor urn models. The construction uses random variables from
the urn
model which are different from the random variables for which central
limit
theorems are available in the two color case.
http://front.math.ucdavis.edu/math.PR/0512325
---------------------------------------------------------------
3884. COUPLING ALL THE LEVY STOCHASTIC AREAS OF MULTIDIMENSIONAL
BROWNIAN MOTION
Wilfrid Kendall
It is shown how to construct a successful co-adapted coupling of two
copies
of an n-dimensional Brownian motion while simultaneously coupling all
corresponding copies of Levy stochastic areas. It is conjectured that
successful co-adapted couplings still exist when the Levy stochastic
areas are
replaced by a finite set of multiply-iterated path-and-time
integrals, subject
to algebraic compatibility of the initial conditions.
http://front.math.ucdavis.edu/math.PR/0512336
---------------------------------------------------------------
3885. QUANTUM STOCHASTIC SEMIGROUPS AND THEIR GENERATORS
V. P. Belavkin
A rigged Hilbert space characterisation of the unbounded generators of
quantum completely positive (CP) stochastic semigroups is given. The
general
form and the dilation of the stochastic completely dissipative (CD)
equation
over the algebra L(H) is described, as well as the unitary quantum
stochastic
dilation of the subfiltering and contractive flows with unbounded
generators is
constructed.
http://front.math.ucdavis.edu/math.PR/0512360
---------------------------------------------------------------
3886. QUANTUM STOCHASTIC CALCULUS AND QUANTUM NONLINEAR FILTERING
V. P. Belavkin
A *-algebraic indefinite structure of quantum stochastic (QS)
calculus is
introduced and a continuity property of generalized nonadapted QS
integrals is
proved under the natural integrability conditions in an infinitely
dimensional
nuclear space. The class of nondemolition output QS processes in
quantum open
systems is characterized in terms of the QS calculus, and the problem
of QS
nonlinear filtering with respect to nondemolition continuous
measurments is
investigated. The stochastic calculus of a posteriori conditional
expectations
in quantum observed systems is developed and a general quantum filtering
stochastic equation for a QS process is derived. An application to the
description of the spontaneous collapse of the quantum spin under
continuous
observation is given.
http://front.math.ucdavis.edu/math.PR/0512362
---------------------------------------------------------------
3887. LOGARITHMIC ASYMPTOTICS FOR THE NUMBER OF PERIODIC ORBITS OF
THE TEICHMUELLER FLOW ON VEECH'S SPACE OF ZIPPERED RECTANGLES
Alexander I. Bufetov
The logarithmic asymptotics is computed for the growth of the number of
periodic orbits for the Teichmueller flow on Veech's moduli space of
zippered
rectangles. The rate is equal to the entropy of the flow with respect
to the
absolutely continuous invariant measure.
http://front.math.ucdavis.edu/math.DS/0511035
---------------------------------------------------------------
3888. LOCALIZATION TRANSITION FOR A COPOLYMER IN AN EMULSION
F den Hollander and S G Whittington
In this paper we study a two-dimensional directed self-avoiding walk
model of
a random copolymer in a random emulsion.
http://front.math.ucdavis.edu/math.PR/0512374
---------------------------------------------------------------
3889. GIBBS DISTRIBUTIONS FOR RANDOM PARTITIONS GENERATED BY A
FRAGMENTATION PROCESS
Nathanael Berestycki (U.B.C.) and Jim Pitman (U.C. BERKELEY)
In this paper we study random partitions of {1,...,n} where every
cluster of
size j can be in any of w(j) possible internal states. The Gibbs (n,k,w)
distribution is obtained by sampling uniformly among such partitions
with k
clusters. Gibbs distributions arise naturally as equilibrium
distributions of
reversible coagulation - fragmentation processes. The goal of this
work is to
study random processes where at step k the process has the Gibbs (n,k,w)
distribution, so that this microscopical equilibrium is subject to
irreversible
fragmentation as time evolves. It is not always possible to combine
those two
features, and in our main result we identify those weight sequences w
(j) for
which such a process exists subject to some simplifying assumptions.
In this
case the time-reversed process turns out to be the discrete Marcus-
Lushnikov
coalescent process with affine collision rate K(x,y)=a+b(x+y) for
some real
numbers a and b.
http://front.math.ucdavis.edu/math.PR/0512378
---------------------------------------------------------------
3890. A QUANTITATIVE INVESTIGATION INTO THE ACCUMULATION OF ROUNDING
ERRORS IN THE NUMERICAL SOLUTION OF ODES
Sebastian Mosbach and Amanda G. Turner
We examine numerical rounding errors of some deterministic solvers for
systems of ordinary differential equations (ODEs). We show that the
accumulation of rounding errors results in a solution that is
inherently random
and we obtain the theoretical distribution of the trajectory as a
function of
time, the step size and the numerical precision of the computer. We
consider,
in particular, systems which amplify the effect of the rounding
errors so that
over long time periods the solutions exhibit divergent behaviour. By
performing
multiple repetitions with different values of the time step size, we
observe
numerically the random distributions predicted theoretically. We
mainly focus
on the explicit Euler and RK4 methods but also briefly consider more
complex
algorithms such as the implicit solvers VODE and RADAU5.
http://front.math.ucdavis.edu/math.NA/0512364
---------------------------------------------------------------
3891. NORMAL DOMINATION OF (SUPER)MARTINGALES
Iosif Pinelis
Let (S_0,S_1,...) be a supermartingale relative to a nondecreasing
sequence
of \sigma-algebras (H_{\le0},H_{\le1},...), with S_0\le0 almost
surely (a.s.)
and differences X_i:=S_i-S_{i-1}. Suppose that for every i=1,2,...
there exist
H_{\le(i-1)}-measurable r.v.'s C_{i-1} and D_{i-1} and a positive
real number
s_i such that C_{i-1}\le X_i\le D_{i-1} and D_{i-1}-C_{i-1}\le 2 s_i
a.s. Then
for all real t and natural n one has \E f_t(S_n)\le\E f_t(sZ), where
f_t(x):=\max(0,x-t)^5, s:=\sqrt{s_1^2+...+s_n^2}, and Z is N(0,1). In
particular, this implies P(S_n\ge x)\le c_{5,0}P(Z\ge x/s) for all x
in \R,
where c_{5,0}=5!(e/5)^5=5.699.... Results for \max_{0\le k\le n}S_k
in place of
S_n and for concentration of measure also follow.
http://front.math.ucdavis.edu/math.PR/0512382
---------------------------------------------------------------
3892. RELATIVE ENTROPY AND WAITING TIMES FOR CONTINUOUS-TIME MARKOV
PROCESSES
Jean-Rene Chazottes and Cristian Giardina and Frank Redig
For discrete-time stochastic processes, there is a close connection
between
return/waiting times and entropy. Such a connection cannot be
straightforwardly
extended to the continuous-time setting. Contrarily to the discrete-
time case
one does need a reference measure and so the natural object is
relative entropy
rather than entropy. In this paper we elaborate on this in the case of
continuous-time Markov processes with finite state space. A reference
measure
of special interest is the one associated to the time-reversed
process. In that
case relative entropy is interpreted as the entropy production rate.
The main
results of this paper are: almost-sure convergence to relative
entropy of
suitable waiting-times and their fluctuation properties (central
limit theorem
and large deviation principle).
http://front.math.ucdavis.edu/math.PR/0512386
---------------------------------------------------------------
3893. ASYMPTOTIC DIRECTION FOR RANDOM WALKS IN RANDOM ENVIRONMENTS
Fran\c{c}ois Simenhaus (PMA)
In this paper we study the property of asymptotic direction for
random walks
in random i.i.d. environments (RWRE). We prove that if the set of
directions
where the walk is transient is non empty and open, the walk admits an
asymptotic direction. The main tool to obtain this result is the
construction
of a renewal structure with cones. We also prove that RWRE admits at
most two
opposite asymptotic directions.
http://front.math.ucdavis.edu/math.PR/0512388
---------------------------------------------------------------
3894. RANDOMLY GROWING BRAID ON THREE STRANDS AND THE MANTA RAY, WITH
APPENDIX
Jean Mairesse and Fr\'ed\'eric Math\'eus
Consider the braid group B3 = < a,b | aba = bab > and the nearest
neighbor
random walk defined by a probability \nu with support
{a,b,a^-1,b^-1}. The rate
of escape of the walk is explicitely expressed in function of the unique
solution of a set of eight polynomial equations of degree three over
eight
indeterminates. We also explicitely describe the harmonic measure of the
induced random walk on B3 quotiented by its center. The method and
results
apply, mutatis mutandis, to nearest neighbor random walks on dihedral
Artin
groups.
http://front.math.ucdavis.edu/math.PR/0512391
---------------------------------------------------------------
3895. A UNIVERSAL DILATION OF DISCRETE MARKOV EVOLUTIONS
M. Gregoratti
Given a finite state space E, we build a universal dilation for all
possible
discrete time Markov chains on E, homogeneous or not: we introduce a
second
system (an ``environment'') and a deterministic invertible time-
homogeneous
global evolution of the system E with this environment such that any
Markov
evolution of E can be realized by a proper choice of the initial
(random) state
of the environment, which therefore determines the transition
probabilities of
the system. We also compare this dilation with the quantum dilations
of a
Quantum Dynamical Semigroup: given a Classical Markov Semigroup, we
show that
it can be extended to a Quantum Dynamical Semigroup for which we can
find a
quantum dilation to a group of *-automorphisms admitting an invariant
abelian
subalgebra where this quantum dilation gives just our classical
dilation.
http://front.math.ucdavis.edu/math.PR/0512393
---------------------------------------------------------------
3896. LARGE DEVIATIONS OF THE EMPIRICAL CURRENT IN INTERACTING
PARTICLE SYSTEMS
L. Bertini and A. De Sole and D. Gabrielli and G. Jona-Lasinio
and C. Landim
We study current fluctuations in lattice gases in the hydrodynamic
scaling
limit. More precisely, we prove a large deviation principle for the
empirical
current in the symmetric simple exclusion process with rate
functional I. We
then estimate the asymptotic probability of a fluctuation of the average
current over a large time interval and show that the corresponding rate
function can be obtained by solving a variational problem for the
functional I.
For the symmetric simple exclusion process the minimizer is time
independent so
that this variational problem can be reduced to a time independent
one. On the
other hand, for other models the minimizer is time dependent. This
phenomenon
is naturally interpreted as a dynamical phase transition.
http://front.math.ucdavis.edu/math.PR/0512394
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3897. CONFORMAL INVARIANCE OF ISORADIAL DIMER MODELS & THE CASE OF
TRIANGULAR QUADRI-TILINGS
B. de Tili\`ere
We consider dimer models on graphs which are bipartite, periodic and
satisfy
a geometric condition called {\em isoradiality}, defined in \cite
{Kenyon3}. We
show that the scaling limit of the height function of any such dimer
model is
$1/\sqrt{\pi}$ times a Gaussian free field. Triangular quadri-tilings
were
introduced in \cite{Bea}; they are dimer models on a family of
isoradial graphs
arising form rhombus tilings. By means of two height functions, they
can be
interpreted as random interfaces in dimension 2+2. We show that the
scaling
limit of each of the two height functions is $1/\sqrt{\pi}$ times a
Gaussian
free field, and that the two Gaussian free fields are independent.
http://front.math.ucdavis.edu/math.PR/0512395
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3898. THE MONOTONICITY CONDITION FOR BSDE ON MANIFOLDS
Fabrice Blache (IAM)
In two preceding articles, we studied the problem of the existence and
uniqueness of a solution to some general BSDE on manifolds. In these two
articles, we assumed some Lipschitz conditions on the drift $f(b,x,z)
$. The
purpose of this article is to extend the existence and uniqueness
results under
weaker assumptions, in particular a monotonicity condition in the
variable $x$.
This extends well-known results for Euclidean BSDE.
http://front.math.ucdavis.edu/math.PR/0512403
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3899. OPERATOR MARKOVIAN COCYCLES VIA ASSOCIATED SEMIGROUPS
J. Martin Lindsay and Stephen J. Wills
A recent characterisation of Fock-adapted contraction operator
stochastic
cocycles on a Hilbert space, in terms of their associated semigroups,
yields a
general principle for the construction of such cocycles by
approximation of
their stochastic generators. This leads to new existence results for
quantum
stochastic differential equations. We also give necessary and sufficient
conditions for a cocycle to satisfy such an equation.
http://front.math.ucdavis.edu/math.FA/0512398
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3900. MARKOV MEASURES ON YOUNG TABLEAUX AND INDUCED REPRESENTATIONS
ON THE INFINITE SYMMETRIC GROUP
A.M.Vershik and N.V.Tsilevich
We show that the class of inductive limits of the representations of
finite
symmetric groups with simple spectrum coinsides with the class of Markov
representations of the infinite symmetric group associated with
Markov measures
on the space of infinite Young tableaux.
We also show that the representations of infinite symmetric group
induced
from identity representation of two-block Young subgroup are Markov
representations and find explicit formulas for transition
probabilities of
corresponding Markov measure on the Young diagrmas.
Induced two-row representations of finite symmetric group are
studied using
tensor model of those representations which alows easily to obtain
the formulas
for Gel'fand-Zetlin basis.
http://front.math.ucdavis.edu/math.RT/0512389
---------------------------------------------------------------
3901. RECONSTRUCTION THEOREM FOR QUANTUM STOCHASTIC PROCESSES
V. P. Belavkin
Statistically interpretable axioms are formulated that define a quantum
stochastic process (QSP) as a causally ordered operator field in an
arbitrary
space-time region T of an open quantum system under a sequential
observation at
a discrete space-time localization. It is shown that to every QSP
described in
the weak sense by a self-consistent system of causally ordered
correlation
kernels there corresponds a unique, up to unitary equivalence,
minimal QSP in
the strong sense. It is shown that the proposed QSP construction,
which reduces
in the case of the linearly ordered discrete T=Z to the construction
of the
inductive limit of Lindblad's canonical representations, corresponds to
Kolmogorov's classical reconstruction if the order on T is ignored
and leads to
Lewis construction if one uses the system of all (not only causal)
correlation
kernels, regarding this system as lexicographically preordered on T. The
approach presented encompasses both nonrelativistic and relativistic
irreversible dynamics of open quantum systems and fields satisfying the
conditions of local commutativity and semigroup covariance. Also
given are
necessary and sufficient conditions of dynamicity (or conditional
Markovianity)
and regularity, these leading to the properties of complete mixing
(relaxation)
and ergodicity of the QSP.
http://front.math.ucdavis.edu/math.PR/0512410
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3902. SEMILOGICS, QUASILOGICS AND OTHER QUANTUM STRUCTURES
V. P. Belavkin
We give an axiomatic formulation of quantum structures like
semilogics and
quasilogics which generalize the boolean semirings of events and
fuzzy logics.
The notions of distributions, states, representations observables and
semiobservables are introduced and their Hilbert space realizations
are found.
The closed and open structures in semilogics are introduced and the
regular
distributions on the semilogics are studied.
http://front.math.ucdavis.edu/math.PR/0512413
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3903. OCCUPATION TIME FLUCTUATIONS OF POISSON AND EQUILIBRIUM FINITE
VARIANCE BRANCHING SYSTEMS
Piotr Milos
Functional limit theorems are presented for the rescaled occupation time
fluctuations process of a critical finite variance branching particle
system in
$R^d$ with symmetric a-stable motion starting off from either a standard
Poisson random field or from the equilibrium distribution for
intermediate
dimensions a<d<2a. The limit processes are determined sub-fractional and
fractional Brownian motion respectively.
http://front.math.ucdavis.edu/math.PR/0512414
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3904. QUANTUM PROBABILITIES AND PARADOXES OF THE QUANTUM CENTURY
V. P. Belavkin
A history and drama of the development of quantum probability theory is
outlined starting from the discovery of the Plank's constant exactly
a 100
years ago. It is shown that before the rise of quantum mechanics 75
years ago,
the quantum theory had appeared first in the form of the statistics
of quantum
thermal noise and quantum spontaneous jumps which have never been
explained by
quantum mechanics. Moreover, the only reasonable probabilistic
interpretation
of quantum theory put forward by Max Born was in fact in irreconcilable
contradiction with traditional mechanical reality and classical
probabilistic
causality. This led to numerous quantum paradoxes, some of them due
to the
great inventors of quantum theory such as Einstein and Schroedinger.
They are
reconsidered in this paper from the modern quantum probabilistic
point of view.
http://front.math.ucdavis.edu/math.PR/0512415
---------------------------------------------------------------
3905. GENERALIZED PROBABILITIES TAKING VALUES IN NON-ARCHIMEDEAN
FIELDS AND TOPOLOGICAL GROUPS
Andrei Khrennikov
We develop an analogue of probability theory for probabilities taking
values
in topological groups. We generalize Kolmogorov's method of
axiomatization of
probability theory: main distinguishing features of frequency
probabilities are
taken as axioms in the measure-theoretic approach. We also present a
review of
non-Kolmogorovian probabilistic models including models with
negative, complex,
and $p$-adic valued probabilities. The latter model is discussed in
details.
The introduction of $p$-adic (as well as more general non-Archimedean)
probabilities is one of the main motivations for consideration of
generalized
probabilities taking values in topological groups which are distinct
from the
field of real numbers. We discuss applications of non-Kolmogorovian
models in
physics and cognitive sciences. An important part of this paper is
devoted to
statistical interpretation of probabilities taking values in
topological groups
(and in particular in non-Archimedean fields).
http://front.math.ucdavis.edu/math.PR/0512427
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3906. ON MAXIMUM INCREASE AND DECREASE OF BROWNIAN MOTION
Paavo Salminen and Pierre Vallois
The joint distribution of maximum increase and decrease for Brownian
motion
up to an independent exponential time is computed. This is achieved by
decomposing the Brownian path at the hitting times of the infimum and
the
supremum before the exponential time. It is seen that an important
element in
our formula is the distribution of the maximum decrease for the three
dimensional Bessel process with drift started from 0 and stopped at
the first
hitting of a given level. From the joint distribution of the maximum
increase
and decrease it is possible to calculate the correlation coefficient
between
these at a fixed time and this is seen to be -0.47936... .
http://front.math.ucdavis.edu/math.PR/0512440
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3907. A STOCHASTIC LAGRANGIAN REPRESENTATION OF THE 3-DIMENSIONAL
INCOMPRESSIBLE NAVIER-STOKES EQUATIONS
Peter Constantin and Gautam Iyer
In this paper we derive a representation of the deterministic 3-
dimensional
Navier-Stokes equations based on stochastic Lagrangian paths. The
particle
trajectories obey SDEs driven by a uniform Wiener process; the
inviscid Weber
formula for the Euler equations of ideal fluids is used to recover
the velocity
field. This method admits a self-contained proof of local existence
for the
nonlinear stochastic system, and can be extended to formulate stochastic
representations of related hydrodynamic-type equations, including
viscous
Burgers equations and LANS-alpha models.
http://front.math.ucdavis.edu/math.PR/0511067
---------------------------------------------------------------
3908. CONTROLLED DIFFUSION PROCESSES
Vivek S. Borkar
This article gives an overview of the developments in controlled
diffusion
processes, emphasizing key results regarding existence of optimal
controls and
their characterization via dynamic programming for a variety of cost
criteria
and structural assumptions. Stochastic maximum principle and control
under
partial observations (equivalently, control of nonlinear filters) are
also
discussed. Several other related topics are briefly sketched.
http://front.math.ucdavis.edu/math.PR/0511077
---------------------------------------------------------------
3909. BASIC PROPERTIES OF STRONG MIXING CONDITIONS. A SURVEY AND SOME
OPEN QUESTIONS
Richard C. Bradley
This is an update of, and a supplement to, a 1986 survey paper by the
author
on basic properties of strong mixing conditions.
http://front.math.ucdavis.edu/math.PR/0511078
---------------------------------------------------------------
3910. ASYMPTOTIC ANALYSIS FOR THE RATIO OF THE RANDOM SUM OF SQUARES
TO THE SQUARE OF THE RANDOM SUM WITH APPLICATIONS TO RISK MEASURES
S.A. Ladoucette and J.L. Teugels
Let \{X_1, X_2, ...\} be a sequence of independent and identically
distributed positive random variables of Pareto-type with index
\alpha>0 and
let \{N(t); t\geq 0\} be a counting process independent of the X_i's.
For any
fixed t\geq 0, define T_{N(t)}:=\frac{X_1^2 + X_2^2 + ... + X_{N(t)}
^2} {(X_1 +
X_2 + ... + X_{N(t)})^2} if N(t)\geq 1 and T_{N(t)}:=0 otherwise. We
derive
limiting distributions for T_{N(t)} by assuming some convergence
properties for
the counting process. This is even achieved when both the numerator
and the
denominator defining T_{N(t)} exhibit an erratic behavior
(\mathbb{E}X_1=\infty) or when only the numerator has an erratic
behavior
(\mathbb{E}X_1<\infty and \mathbb{E}X_1^2=\infty). Thanks to these
results, we
obtain asymptotic properties pertaining to both the sample
coefficient of
variation and the sample dispersion.
http://front.math.ucdavis.edu/math.PR/0511082
---------------------------------------------------------------
3911. PLONGEMENT STOCHASTIQUE DES SYST\`{E}MES LAGRANGIENS
Jacky Cresson (LM-Besan\c{c}on) and S\'{e}bastien Darses (LM-Besan\c
{c}on)
We define an operator which extends classical differentiation from
smooth
deterministic functions to certain stochastic processes. Based on this
operator, we define a procedure which associates a stochastic analog to
standard differential operators and ordinary differential equations.
We call
this procedure stochastic embedding. By embedding lagrangian systems,
we obtain
a stochastic Euler-Lagrange equation which, in the case of natural
lagrangian
systems, is called the embedded Newton equation. This equation
contains the
stochastic Newton equation introduced by Nelson in his dynamical
theory of
brownian diffusions. Finally, we consider a diffusion with a gradient
drift, a
constant diffusion coefficient and having a probability density
function. We
prove that a necessary condition for this diffusion to solve the
embedded
Newton equation is that its density be the square of the modulus of a
wave
function solution of a linear Schr\"{o}dinger equation.
http://front.math.ucdavis.edu/math.PR/0510655
---------------------------------------------------------------
3912. A MOMENTUM CONSERVING MODEL WITH ANOMALOUS THERMAL CONDUCTIVITY
IN LOW DIMENSION
Giada Basile (CEREMADE) and Cedric Bernardin (UMPA-ENSL) and
Stefano Olla (CEREMADE)
Anomalous large thermal conductivity has been observed numerically and
experimentally in one and two dimensional systems. All explicitly
solvable
microscopic models proposed until now did not explain this phenomenon
and there
is an open debate about the role of conservation of momentum. We
introduce a
model whose thermal conductivity diverges in dimension 1 and 2, while it
remains finite in dimension 3. We compute the finite-size thermal
conductivity
of a system of harmonic oscillators perturbed by a non-linear stochastic
dynamics conserving momentum and energy. In the limit as the size N
of the
system goes to infinity, conductivity diverges like N in dimension 1
and like
ln N in dimension 2. Conductivity remains finite if d=3 or if a
pinning (on
site potential) is present. This result clarify the role of
conservation of
momentum in the anomalous thermal conductivity.
http://front.math.ucdavis.edu/cond-mat/0509688
---------------------------------------------------------------
3913. MISMATCHED CODEBOOKS AND THE ROLE OF ENTROPY-CODING IN LOSSY
DATA COMPRESSION
Ioannis Kontoyiannis (Athens U of Econ & Business) and Rami Zamir
(Tel-Aviv University)
We introduce a universal quantization scheme based on random coding,
and we
analyze its performance. This scheme consists of a source-independent
random
codebook (typically_mismatched_ to the source distribution), followed by
optimal entropy-coding that is_matched_ to the quantized codeword
distribution.
A single-letter formula is derived for the rate achieved by this
scheme at a
given distortion, in the limit of large codebook dimension. The rate
reduction
due to entropy-coding is quantified, and it is shown that it can be
arbitrarily
large. In the special case of "almost uniform" codebooks (e.g., an
i.i.d.
Gaussian codebook with large variance) and difference distortion
measures, a
novel connection is drawn between the compression achieved by the
present
scheme and the performance of "universal" entropy-coded dithered lattice
quantizers. This connection generalizes the "half-a-bit" bound on the
redundancy of dithered lattice quantizers. Moreover, it demonstrates
a strong
notion of universality where a single "almost uniform" codebook is
near-optimal
for_any_ source and_any_ difference distortion measure.
http://front.math.ucdavis.edu/cs.IT/0511009
---------------------------------------------------------------
3914. THE K-CORE AND BRANCHING PROCESSES
Oliver Riordan
The k-core of a graph G is the maximal subgraph of G having minimum
degree at
least k. In 1996, Pittel, Spencer and Wormald found the threshold $
\lambda_c$
for the emergence of a non-trivial k-core in the random graph $G(n,
\lambda/n)$,
and the asymptotic size of the k-core above the threshold. We give a
new proof
of this result using a local coupling of the graph to a suitable
branching
process. This proof extends to a general model of inhomogeneous
random graphs
with independence between the edges. As an example, we study the k-
core in a
certain power-law or `scale-free' graph with a parameter c
controlling the
overall density of edges. For each k at least 3, we find the
threshold value of
c at which the k-core emerges, and the fraction of vertices in the k-
core when
c is \epsilon above the threshold. In contrast to $G(n,\lambda/n)$, this
fraction tends to 0 as \epsilon tends to 0.
http://front.math.ucdavis.edu/math.CO/0511093
---------------------------------------------------------------
3915. LIMITING LAWS FOR LONG BROWNIAN BRIDGES PERTURBED BY THEIR ONE-
SIDED MAXIMUM, III
Bernard Roynette (IEC) and Pierre Vallois (IEC) and Marc Yor (PMA)
Results of penalization of a one-dimensional Brownian motion $(X_t)
$, by its
one-sided maximum $\dis (S_t=\sup_{0 \leq u \leq t}X_u)$, which were
recently
obtained by the authors are improved with the consideration-in the
present
paper- of the asymptotic behaviour of the likewise penalized Brownian
bridges
of length $t$, as $t\to \infty$, or penalizations by functions of $
(S_t,X_t)$,
and also the study of the speed of convergence, as $t\to \infty$, of the
penalized distributions at time $t$.
http://front.math.ucdavis.edu/math.PR/0511102
---------------------------------------------------------------
3916. EXERCISE REGIONS AND CONTINUITY CORRECTIONS FOR (PERPETUAL)
AMERICAN AND BERMUDAN OPTIONS ON MULTIPLE ASSETS
Frederik S Herzberg
In a general Markovian martingale framework for multi-dimensional
options,
the existence of optimal exercise regions for multi-dimensional Bermudan
options is established. Afterwards one can proceed to prove explicit
formulae
and asymptotic results on the perpetual American-Bermudan (barrier)
put option
price difference (``continuity correction'') when the argument of
this function
-- taken to be the (logarithmic) start price -- approaches the exercise
boundary. In particular, results of Feller's shall be generalised to
show that
an extrapolation from the exact Bermudan prices to the American price
cannot be
polynomial in the exercise mesh size in the setting of many common
market
models, and more specific bounds on the natural scaling exponent of the
non-polynomial extrapolation for a number of (both one- and multi-
dimensional)
market models will be deduced.
http://front.math.ucdavis.edu/math.PR/0511106
---------------------------------------------------------------
3917. RUIN ANALYSIS IN CONSTANT ELASTICITY OF VARIANCE MODEL WITH
LARGE INITIAL FUNDS
F. Klebaner and R. Liptser
We consider the value process described by the Constant Elasticity of
Variance Model (CEV), given by the stochastic differential equation $$
dX_t=\alpha X_tdt+\sigma X^\gamma_tdB_t, $$ with $X_0=K$, and $1/2\le
\gamma<1$. Denote the time of ruin $\tau_K=\inf\{t:X_t=0\}$. We give an
asymptotic for the ruin probability by time $T$, $\mathsf{P}(\tau_K
\le T)$
\begin{gather*} \lim\limits_{K\to\infty}
\frac{1}{K^{2(1-\gamma)}}\log\mathsf{P}(\tau_K\le T) =-\begin{cases}
\frac{\alpha}{\sigma^2[1-e^{-2\alpha(1-\gamma)T}]}, & \alpha\ne 0 \
\frac{1}{2\sigma^2(1-\gamma)T}, & \alpha=0 \end{cases}. \end{gather*}
The most
likely paths to ruin is also found. The results are obtained by
solving a
control problem arising with help the Large Deviations Principle (LDP).
http://front.math.ucdavis.edu/math.PR/0511116
---------------------------------------------------------------
3918. LEMME DE COHERENCE ET TH\'{E}OR\`{E}ME DE NOETHER STOCHASTIQUE
Jacky Cresson (LM-Besan\c{c}on) and S\'{e}bastien Darses (LM-Besan\c
{c}on)
The stochastic embedding procedure associates a stochastic Euler-
Lagrange
equation (SEL) to the standard Euler-Lagrange equation (EL). Can we
derive
(SEL) from a generalized least action principle? To address this
question, we
develop a stochastic calculus of variation initiated by Yasue. We give a
stochastic analog F of the lagrangian action functional. We introduce
a notion
of stationarity according to which the solutions of (SEL) are the
stationary
points of F. This notion of stationarity brings coherence to stochastic
calculus of variation with respect to stochastic embedding. Finally,
we prove a
stochastic Noether theorem which introduces an original notion of
stochastic
first integral.
http://front.math.ucdavis.edu/math.PR/0510656
---------------------------------------------------------------
3919. DENSITY OF PATHS OF ITERATED L\'{E}VY TRANSFORMS OF BROWNIAN
MOTION
Marc Malric (PMA)
The L\'{e}vy transform of a Brownian motion B is the Brownian motion
B't, the
integral over (O,t) of sign of Bs with respect to dBs. Call T the
corresponding
transformation on the Wiener space W. We establish that a.s. the
orbit of w in
W under T is dense in W for the compact uniform convergence topology.
http://front.math.ucdavis.edu/math.PR/0511154
---------------------------------------------------------------
3920. JOINT DENSITY FOR THE LOCAL TIMES OF CONTINUOUS-TIME MARKOV CHAINS
D. Brydges and R. van der Hofstad and W. Konig
We investigate the local times of a continuous-time Markov chain on an
arbitrary discrete state space. For fixed finite range of the Markov
chain, we
derive an explicit formula for the joint density of all local times
on the
range, at any fixed time. We use standard tools from the theory of
stochastic
processes and finite-dimensional complex calculus. We apply this
formula in the
following directions: (1) we derive large deviation upper estimates
for the
normalized local times beyond the exponential scale, (2) we derive
the upper
bound in Varadhan's Lemma for any measurable functional of the local
times, (3)
we derive large deviation upper bounds for continuous-time simple
random walk
on large subboxes of $\Z^d$ tending to $\Z^d$ as time diverges, and
(4) we
prove the analog of the well-known Ray-Knight description of Brownian
local
times for any nearest-neighbor continuous-time Markov chain on $\Z$,
with
particularly explicit formulas for simple random walk.
http://front.math.ucdavis.edu/math.PR/0511169
---------------------------------------------------------------
3921. REGENERATIVE REAL TREES
Mathilde Weill (DMA)
In this work, we give a description of all sigma-finite measures on
the space
of rooted compact real trees which satisfy a certain regenerative
property. We
show that any infinite measure which satisfies the regenerative
property is the
"law" of a Levy tree, that is, the "law" of a tree-valued random
variable that
describes the genealogy of a population evolving according to a
continuous-state branching process. On the other hand, we prove that a
probability measure with the regenerative property must be the law of
the
genealogical tree associated with a continuous-time discrete-state
branching
process.
http://front.math.ucdavis.edu/math.PR/0511172
---------------------------------------------------------------
3922. PERCOLATION FOR THE STABLE MARRIAGE OF POISSON AND LEBESGUE
Marcelo Ventura Freire and Serguei Popov and Marina Vachkovskaia
Let $\Xi$ be the set of points (we call the elements of $\Xi$
centers) of
Poisson point process in ${\bf R}^d$, $d\geq 2$, with unit intensity.
Consider
the allocation of ${\bf R}^d$ to $\Xi$ which is stable in the sense of
Gale-Shapley marriage problem and in which each center claims a
region of
volume $\alpha\leq 1$. We prove that there is no percolation in the
set of
claimed sites if $\alpha$ is small enough, and that, for high
dimensions, there
is percolation in the set of claimed sites if $\alpha<1$ is large
enough.
http://front.math.ucdavis.edu/math.PR/0511186
---------------------------------------------------------------
3923. HARRIS PROCESSES
S Sherly and M K Jose and E Sandhya and N Raju
In this paper, we develop two stochastic models where the variable under
consideration follows Harris distribution. The mean and variance of the
processes are derived and the processes are shown to be non-
stationary. In the
second model, starting with a Poisson process, an alternate way of
obtaining
Harris process is introduced.
http://front.math.ucdavis.edu/math.PR/0510658
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3924. METRIC CONSTRUCTION, STOPPING TIMES AND PATH COUPLING
Magnus Bordewich and Martin Dyer and Marek Karpinski
In this paper we examine the importance of the choice of metric in path
coupling, and the relationship of this to \emph{stopping time
analysis}. We
give strong evidence that stopping time analysis is no more powerful
than
standard path coupling. In particular, we prove a stronger theorem
for path
coupling with stopping times, using a metric which allows us to restrict
analysis to standard one-step path coupling. This approach provides
insight for
the design of non-standard metrics giving improvements in the
analysis of
specific problems.
We give illustrative applications to hypergraph independent sets
and SAT
instances, hypergraph colourings and colourings of bipartite graphs.
http://front.math.ucdavis.edu/math.PR/0511202
---------------------------------------------------------------
3925. A NECESSARY AND SUFFICIENT CONDITION FOR THE TAIL-TRIVIALITY OF
A RECURSIVE TREE PROCESS
Antar Bandyopadhyay
Given a recursive distributional equation (RDE) and a solution $\mu$
of it,
we consider the tree indexed invariant process called the recursive tree
process (RTP) with marginal $\mu$. We introduce a new type of bivariate
uniqueness property which is different from the one defined by Aldous
and
Bandyopadhyay (2005), and we prove that this property is equivalent to
tail-triviality for the RTP. Thus obtaining a necessary and sufficient
condition to determine tail-triviality for a RTP in general. As an
application
we consider Aldous' (2000) construction of the frozen percolation
process on a
infinite regular tree and show that the associated RTP has a trivial
tail.
http://front.math.ucdavis.edu/math.PR/0511203
---------------------------------------------------------------
3926. HIGH-RESOLUTION PRODUCT QUANTIZATION FOR GAUSSIAN PROCESSES
UNDER SUP-NORM DISTORTION
Harald Luschgy and Gilles Pag\`{e}s (PMA)
We derive high-resolution upper bounds for optimal product
quantization of
pathwise contionuous Gaussian processes respective to the supremum
norm on
[0,T]^d. Moreover, we describe a product quantization design which
attains this
bound. This is achieved under very general assumptions on random series
expansions of the process. It turns out that product quantization is
asymptotically only slightly worse than optimal functional
quantization. The
results are applied e.g. to fractional Brownian sheets and the
Ornstein-Uhlenbeck process.
http://front.math.ucdavis.edu/math.PR/0511208
---------------------------------------------------------------
3927. INVERSE LITTLEWOOD-OFFORD THEOREMS AND THE CONDITION NUMBER OF
RANDOM DISCRETE MATRICES
Terence Tao and Van Vu
Consider a random sum $\eta_1 v_1 + ... + \eta_n v_n$, where
$\eta_1,...,\eta_n$ are i.i.d. random signs and $v_1,...,v_n$ are
integers. The
Littlewood-Offord problem asks to maximize concentration
probabilities such as
$\P(\eta_1 v_1 + ... + \eta_n v_n = 0)$ subject to various hypotheses
on the
$v_1,...,v_n$. In this paper we develop an \emph{inverse} Littlewood-
Offord
theorem (somewhat in the spirit of Freiman's inverse sumset theorem),
which
starts with the hypothesis that a concentration probability is large,
and
concludes that almost all of the $v_1,...,v_n$ are efficiently
contained in an
arithmetic progression. As an application we give some new bounds on the
distribution of the least singular value of a random Bernoulli
matrix, which in
turn gives upper tail estimates on the condition number.
http://front.math.ucdavis.edu/math.PR/0511215
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3928. ON THE SEPARATION PRINCIPLE OF QUANTUM CONTROL
Luc Bouten and Ramon van Handel
It is well known that continuous quantum measurements and nonlinear
filtering
can be developed within the framework of the quantum stochastic
calculus of
Hudson-Parthasarathy. The addition of real-time feedback control has
been
discussed by many authors, but never in a rigorous way. Here we
introduce the
notion of a controlled quantum flow, where feedback is taken into
account by
allowing the coefficients of the quantum stochastic differential
equation to be
adapted processes in the observation algebra. We then prove a separation
theorem for quantum control: the admissible control that minimizes a
given cost
function is only a function of the filter, provided that the
associated Bellman
equation has a sufficiently regular solution. Along the way we obtain
results
on the innovations problem in the quantum setting.
http://front.math.ucdavis.edu/math-ph/0511021
---------------------------------------------------------------
3929. NONLINEARITY, CORRELATION AND THE VALUATION OF EMPLOYEE STOCK
OPTIONS
M. R. Grasselli
We propose a discrete time algorithm for the valuation of employee stock
options based on exponential indifference prices and taking into
account both
the possibility of partial exercise of a fraction of the options and
the use of
a correlated traded asset to hedge part of their risk. We determine
the optimal
exercise policy under this conditions and present numerical results
showing how
both effects can significantly change the value of the option for an
employee,
as well as its cost for the issuing firm.
http://front.math.ucdavis.edu/math.ST/0511234
---------------------------------------------------------------
3930. AVOIDING DEFEAT IN A BALLS-IN-BINS PROCESS WITH FEEDBACK
Roberto Oliveira and Joel Spencer
Imagine that there are two bins to which balls are added
sequentially, and
each incoming ball joins a bin with probability proportional to the p-
th power
of the number of balls already there. A general result says that if
p>1/2,
there almost surely is some bin that will have more balls than the
other at all
large enough times, a property that we call eventual leadership.
In this paper, we compute the asymptotics of the probability that
bin 1
eventually leads when the total initial number of balls $t$ is large
and bin 1
has a fraction \alpha<1/2 of the balls; in fact, this probability is
\exp(c_p(\alpha)t + O{t^{2/3}}) for some smooth, strictly negative
function
c_p. Moreover, we show that conditioned on this unlikely event, the
fraction of
balls in the first bin can be well-approximated by the solution to a
certain
ordinary differential equation.
http://front.math.ucdavis.edu/math.PR/0510663
---------------------------------------------------------------
3931. PROBABILITIES ON CLADOGRAMS: INTRODUCTION TO THE ALPHA MODEL
Daniel J. Ford
The alpha model, a parametrized family of probabilities on cladograms
(rooted
binary leaf labeled trees), is introduced. This model is Markovian
self-similar, deletion-stable (sampling consistent), and passes
through the
Yule, Uniform and Comb models. An explicit formula is given to
calculate the
probability of any cladogram or tree shape under the alpha model.
Sackin's and
Colless' index are shown to be $O(n^{1+\alpha})$ with asymptotic
covariance
equal to 1. Thus the expected depth of a random leaf with $n$ leaves is
$O(n^\alpha)$. The number of cherries on a random alpha tree is shown
to be
asymptotically normal with known mean and variance. Finally the shape of
published phylogenies is examined, using trees from Treebase.
http://front.math.ucdavis.edu/math.PR/0511246
---------------------------------------------------------------
3932. FINITE-DIMENSIONAL APPROXIMATION FOR THE DIFFUSION COEFFICIENT
IN SIMPLE EXCLUSION PROCESS
M. D. Jara
We show that for the mean zero simple exclusion process and for the
asymmetric simple exclusion process in dimension d > 2, the self-
diffusion
coefficient of a tagged particle is stable when approximated by simple
exclusion processes on large periodic lattices. The proof relies on a
similar
property for the Sobolev inner product associated to the generator of
the
process.
http://front.math.ucdavis.edu/math.PR/0511249
---------------------------------------------------------------
3933. WEAK LOGARITHMIC SOBOLEV INEQUALITIES AND ENTROPIC CONVERGENCE
Patrick Cattiaux (MODAL'X and CMAP) and Ivan Gentil (CEREMADE) and
Arnaud Guillin (CEREMADE)
In this paper we introduce and study a weakened form of logarithmic
Sobolev
inequalities in connection with various others functional
inequalities (weak
Poincar\'{e} inequalities, general Beckner inequalities...). We also
discuss
the quantitative behaviour of relative entropy along a symmetric
diffusion
semi-group. In particular, we exhibit an example where Poincar\'{e}
inequality
can not be used for deriving entropic convergence whence weak
logarithmic
Sobolev inequality ensures the result.
http://front.math.ucdavis.edu/math.PR/0511255
---------------------------------------------------------------
3934. EXPONENTIAL FUNCTIONALS OF LEVY PROCESSES
Jean Bertoin and Marc Yor
This text surveys properties and applications of the exponential
functional
$\int_0^t\exp(-\xi_s)ds$ of real-valued L\'evy processes $\xi=(\xi_t,t
\geq0)$.
http://front.math.ucdavis.edu/math.PR/0511265
---------------------------------------------------------------
3935. PROBABILITY & INCOMPRESSIBLE NAVIER-STOKES EQUATIONS: AN
OVERVIEW OF SOME RECENT DEVELOPMENTS
Edward C. Waymire
This is largely an attempt to provide probabilists some orientation
to an
important class of non-linear partial differential equations in applied
mathematics, the incompressible Navier-Stokes equations. Particular
focus is
given to the probabilistic framework introduced by LeJan and Sznitman
[Probab.
Theory Related Fields 109 (1997) 343-366] and extended by
Bhattacharya et al.
[Trans. Amer. Math. Soc. 355 (2003) 5003-5040; IMA Vol. Math. Appl.,
vol. 140,
2004, in press]. In particular this is an effort to provide some
foundational
facts about these equations and an overview of some recent results
with an
indication of some new directions for probabilistic consideration.
http://front.math.ucdavis.edu/math.PR/0511266
---------------------------------------------------------------
3936. SOME RECENT ASPECTS OF RANDOM CONFORMALLY INVARIANT SYSTEMS
Wendelin Werner
These are the lecture notes from a course given in July 2005 at the
summer
school in Les Houches. We describe some recent results concerning
two-dimensional conformally invariant systems. In particular, we discuss
conformally invariant measures on loops and conformal loop-ensembles
(CLE).
http://front.math.ucdavis.edu/math.PR/0511268
---------------------------------------------------------------
3937. ON THE ERGODIC PRINCIPLE FOR MARKOV AND QUADRATIC STOCHASTIC
PROCESSES AND ITS RELATIONS
Nasir Ganikhodjaev and Hasan Akin and Farrukh Mukhamedov
In the paper we prove that a quadratic stochastic process satisfies the
ergodic principle if and only if the associated Markov process
satisfies one.
http://front.math.ucdavis.edu/math.PR/0511270
---------------------------------------------------------------
3938. APPROXIMATE MCKEAN-VLASOV REPRESENTATIONS FOR A CLASS OF SPDES
Dan Crisan and Jie Xiong
The solution $\vartheta =(\vartheta_{t})_{t\geq 0}$ of a class of linear
stochastic partial differential equations is approximated using
Clark's robust
representation approach (\cite{c}, \cite{cc}). The ensuing
approximations are
shown to coincide with the time marginals of solutions of a certain
McKean-Vlasov type equation. We prove existence and uniqueness of the
solution
of the McKean-Vlasov equation. The result leads to a representation of
$\vartheta $as a limit of empirical distributions of systems of equally
weighted particles. In particular, the solution of the Zakai equation
and that
of the Kushner-Stratonovitch equation (the two main equations of
nonlinear
filtering) are shown to be approximated the empirical distribution of
systems
of particles that have equal weights (unlike those presented in \cite
{kj1} and
\cite{kj2}) and do not require additional correction procedures (such
as those
introduced in \cite{dan3}, \cite{dan4}, \cite{dmm}, etc).
http://front.math.ucdavis.edu/math.PR/0510668
---------------------------------------------------------------
3939. COMPUTABLE CONVERGENCE RATES FOR SUBGEOMETRICALLY ERGODIC
MARKOV CHAINS
Randal Douc (CMAP) and Eric Moulines (LTCI) and Philippe Soulier
(MODAL'X)
In this paper, we give quantitative bounds on the $f$-total variation
distance from convergence of an Harris recurrent Markov chain on an
arbitrary
under drift and minorisation conditions implying ergodicity at a sub-
geometric
rate. These bounds are then specialized to the stochastically
monotone case,
covering the case where there is no minimal reachable element. The
results are
illustrated on two examples from queueing theory and Markov Chain
Monte Carlo.
http://front.math.ucdavis.edu/math.PR/0511273
---------------------------------------------------------------
3940. ASYMPTOTIC EXPANSION FOR INVERSE MOMENTS OF BINOMIAL AND
POISSON DISTRIBUTIONS
Marko Znidaric
An asymptotic expansion for inverse moments of positive binomial and
Poisson
distributions is derived. The expansion coefficients of the
asymptotic series
are given by the positive central moments of the distribution.
Compared to
previous results, a single expansion formula covers all (also non-
integer)
inverse moments. In addition, the approach can be generalized to
other positive
distributions.
http://front.math.ucdavis.edu/math.ST/0511226
---------------------------------------------------------------
3941. EXISTENCE OF THE ZERO RANGE PROCESS AND A DEPOSITION MODEL
WITH SUPERLINEAR GROWTH RATES
M. Balazs and F. Rassoul-Agha and T. Seppalainen and S. Sethuraman
We give a construction of the totally asymmetric zero range process
and the
so-called bricklayers' process in the attractive case. The novelty is
that we
allow jump rates to grow as fast as exponentially. These processes
have not
been constructed for any jump rate growing faster than linearly. We
also prove
many of the usual semigroup properties, and show that a family of
iid. product
measures, one for each particle density, is invariant and extremal
for the
process. Extremality is proved using a new approach, which is rather
simple
compared to ergodicity proofs found in the literature.
http://front.math.ucdavis.edu/math.PR/0511287
---------------------------------------------------------------
3942. CAPITAL PROCESS AND OPTIMALITY PROPERTIES OF BAYESIAN SKEPTIC
IN THE FAIR AND BIASED COIN GAMES
Masayuki Kumon and Akimichi Takemura and Kei Takeuchi
We study capital process behavior in the fair-coin game and biased-
coin games
in the framework of the game-theoretic probability of Shafer and Vovk
(2001).
We show that if Skeptic uses a Bayesian strategy with a beta prior,
the capital
process is lucidly expressed in terms of the past average of
Reality's moves.
From this it is proved that the Skeptic's Bayesian strategy weakly
forces the
strong law of large numbers (SLLN) with the convergence rate of O
(\sqrt{\log
n/n})$ and if Reality violates SLLN then the exponential growth rate
of the
capital process is very accurately described in terms of the Kullback
divergence between the average of Reality's moves when she violates
SLLN and
the average when she observes SLLN. We also investigate optimality
properties
associated with Bayesian strategy.
http://front.math.ucdavis.edu/math.ST/0510662
---------------------------------------------------------------
3943. INTRODUCTION TO DETERMINANTAL POINT PROCESSES FROM A QUANTUM
PROBABILITY VIEWPOINT
Alex D. Gottlieb
Determinantal point processes on a measure space X whose kernels
represent
trace class Hermitian operators on L^2(X) are associated to
"quasifree" density
operators on the Fock space over L^2(X).
http://front.math.ucdavis.edu/math.PR/0511334
---------------------------------------------------------------
3944. A NOTE ON A.S. FINITENESS OF PERPETUAL INTEGRAL FUNCTIONALS OF
DIFFUSIONS
Paavo Salminen and Marc Yor (PMA)
In this note, with the help of the boundary classification of
diffusions, we
derive a criterion of the convergence of perpetual integral
functionals of
transient real-valued diffusions. In the particular case of transient
Bessel
processes, we note that this criterion agrees with the one obtained via
Jeulin's convergence lemma.
http://front.math.ucdavis.edu/math.PR/0511336
---------------------------------------------------------------
3945. SEQUENTIAL AND ASYNCHRONOUS PROCESSES DRIVEN BY STOCHASTIC OR
QUANTUM GRAMMARS AND THEIR APPLICATION TO GENOMICS: A SURVEY
Dimitri Petritis (IRMAR)
We present the formalism of sequential and asynchronous processes
defined in
terms of random or quantum grammars and argue that these processes have
relevance in genomics. To make the article accessible to the
non-mathematicians, we keep the mathematical exposition as elementary as
possible, focusing on some general ideas behind the formalism and
stating the
implications of the known mathematical results. We close with a set
of open
challenging problems.
http://front.math.ucdavis.edu/math.PR/0511346
---------------------------------------------------------------
3946. NON-TANGENTIAL AND PROBABILISTIC BOUNDARY BEHAVIOR OF
PLURIHARMONIC FUNCTIONS
Steve Tanner
Let $u$ be a pluriharmonic function on the unit ball in $C^n$. I
consider the
relationship between the set of points $L_u$ on the boundary of the
ball at
which $u$ converges non-tangentially, and the set of points $\L_u$ at
which $u$
converges along conditioned Brownian paths. For harmonic funcitons $u
$ of two
variables, the result $L_u = \L_u$ (a.e.) has been known for some
time, as has
a counterexample to the same equality for three variable harmonic
functions. I
extend the $L_u = \L_u$ (a.e.) result to pluriharmonic functions in
arbitrary
dimensions.
http://front.math.ucdavis.edu/math.PR/0511368
---------------------------------------------------------------
3947. SHARP ASYMPTOTIC BEHAVIOR FOR WETTING MODELS IN (1+1)-DIMENSION
Francesco Caravenna and Giambattista Giacomin and Lorenzo Zambotti
We consider continuous and discrete (1+1)-dimensional wetting models
which
undergo a localization/delocalization phase transition. Using a
simple approach
based on Renewal Theory we determine the precise asymptotic behavior
of the
partition function, from which we obtain the scaling limits of the
models and
an explicit construction of the infinite volume measure
(thermodynamic limit)
in all regimes, including the critical one.
http://front.math.ucdavis.edu/math.PR/0511376
---------------------------------------------------------------
3948. LACE EXPANSION FOR THE ISING MODEL
Akira Sakai
The lace expansion has been a powerful tool to investigate mean-field
behavior for various stochastic-geometrical models, such as self-
avoiding walk
and percolation, above their respective upper-critical dimension. In
this
paper, we prove for the first time the lace expansion for the Ising
model,
which is independent of the property of the spin-spin coupling. In the
ferromagnetic case, we provide key propositions to prove that, without
requiring the reflection positivity of the spin-spin coupling, the
two-point
function obeys a Gaussian infrared bound for the nearest-neighbor
model with
d>>4 and for the spread-out model with d>4 and L>>1, as well as that the
critical two-point function exhibits a Gaussian asymptotics for the
spread-out
model with d>4 and L>>1. As a result, these models exhibit the
ferromagnetic
mean-field behavior.
http://front.math.ucdavis.edu/math-ph/0510093
---------------------------------------------------------------
3949. ASYMPTOTICS OF COUNTS OF SMALL COMPONENTS IN RANDOM
COMBINATORIAL STRUCTURES AND MODELS OF COAGULATION-FRAGMENTATION
Boris L. Granovsky
We establish necessary and sufficient conditions for convergence of non
scaled multiplicative measures on the set of partitions. The measures
depict
component spectrums of random structures and the equilibrium of some
models of
statistical mechanics, including stochastic processes of
coagulation-fragmentation. Based on the above result, we show that
the common
belief that interacting groups in mean field models become
independent as the
number of particles goes to infinity, is in general not true.
http://front.math.ucdavis.edu/math.PR/0511381
---------------------------------------------------------------
3950. REMARKS ON SOME LINEAR FRACTIONAL STOCHASTIC EQUATIONS
Ivan Nourdin (PMA) and Ciprian A. Tudor (SAMOS)
Using the multiple stochastic integrals we prove an existence and
uniqueness
result for a linear stochastic equation driven by the fractional
Brownian
motion with any Hurst parameter. We study both the one parameter and two
parameter cases. When the drift is zero, we show that in the one-
parameter case
the solution in an exponential, thus positive, function while in the
two-parameter settings the solution is negative on a non-negligible set.
http://front.math.ucdavis.edu/math.PR/0511383
---------------------------------------------------------------
3951. FRAGMENTATION OF COMPOSITIONS AND INTERVALS
Anne-Laure Basdevant (PMA)
The fragmentation processes of exchangeable partitions have already been
studied by several authors. In this paper, we examine rather
fragmentation of
exchangeable compositions, that means partitions of $\mcn$ where the
order of
the blocks counts. We will prove that such a fragmentation is
bijectively
associated to an interval fragmentation. Using this correspondence,
we then
calculate the Hausdorff dimension of certain random closed set that
arise in
interval fragmentations and we study Ruelle's interval fragmentation.
http://front.math.ucdavis.edu/math.PR/0511388
---------------------------------------------------------------
3952. AN INVARIANCE PRINCIPLE FOR AZ\'{E}MA MARTINGALES
Nathanael Enriquez (PMA)
An invariance principle for Az\'{e}ma martingales is presented as
well as a
new device to construct solutions of Emery's structure equations.
http://front.math.ucdavis.edu/math.PR/0511402
---------------------------------------------------------------
3953. THEORY OF AMALGAMATED LP SPACES IN NONCOMMUTATIVE PROBABILITY
Marius Junge and Javier Parcet
Let $f_1, f_2, ..., f_n$ be a family of independent copies of a given
random
variable $f$ in a probability space $(\Omega, \mathcal{F}, \mu)$.
Then, the
following equivalence of norms holds whenever $1 \le q \le p < \infty$
$$\Big(\int_{\Omega} \Big[ \sum_{k=1}^n |f_k|^q \Big]^{\frac{p}{q}} d
\mu
\Big)^{\frac1p} \sim \max_{r \in \{p,q\}} {n^{\frac1r} \Big(\int_
\Omega |f|^r
d\mu \Big)^{\frac1r}}.$$ We prove a noncommutative analogue of this
inequality
for sums of free random variables over a given von Neumann
subalgebra. This
formulation leads to new classes of noncommutative function spaces
which appear
in quantum probability as square functions, conditioned square
functions and
maximal functions. Our main tools are Rosenthal type inequalities for
free
random variables, noncommutative martingale theory and factorization of
operator-valued analytic functions. This allows us to generalize this
inequality as a result for noncommutative $L_p$ in the category of
operator
spaces. Moreover, the use of free random variables produces the right
formulation for $p=\infty$, which has not a commutative counterpart.
http://front.math.ucdavis.edu/math.OA/0511406
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