[Pas] Probability Abstracts 92
pas at www.economia.unimi.it
pas at www.economia.unimi.it
Fri May 5 09:50:39 CEST 2006
May 5, 2006
Letter 92
Probability Abstract Service
Abstracts from Mar-1-2006 to Apr-28-2006
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4110. LARGE DEVIATION FOR DIFFUSIONS AND HAMILTON--JACOBI EQUATION IN
HILBERT SPACES
Jin Feng
Large deviation for Markov processes can be studied by Hamilton--Jacobi
equation techniques. The method of proof involves three steps: First,
we apply
a nonlinear transform to generators of the Markov processes, and
verify that
limit of the transformed generators exists. Such limit induces a
Hamilton--Jacobi equation. Second, we show that a strong form of
uniqueness
(the comparison principle) holds for the limit equation. Finally, we
verify an
exponential compact containment estimate. The large deviation
principle then
follows from the above three verifications. This paper illustrates
such a
method applied to a class of Hilbert-space-valued small diffusion
processes.
The examples include stochastically perturbed Allen--Cahn, Cahn--
Hilliard PDEs
and a one-dimensional quasilinear PDE with a viscosity term. We prove
the
comparison principle using a variant of the Tataru method. We also
discuss
different notions of viscosity solution in infinite dimensions in
such context.
http://front.math.ucdavis.edu/math.PR/0602655
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4111. PASSAGE OF L\'{E}VY PROCESSES ACROSS POWER LAW BOUNDARIES AT
SMALL TIMES
Jean Bertoin (PMA) and Ronald A. Doney and Ross A. Maller (CMA)
We wish to characterise when a L\'{e}vy process $X\_t$ crosses
boundaries
like $t^\kappa$, $\kappa>0$, in a one or two-sided sense, for small
times $t$;
thus, we enquire when $\limsup\_{t\downarrow 0}|X\_t|/t^{\kappa}$,
$\limsup\_{t\downarrow 0}X\_t/t^{\kappa}$ and/or $\liminf\_{t\downarrow
0}X\_t/t^{\kappa}$ are almost surely (a.s.) finite or infinite.
Necessary and
sufficient conditions are given for these possibilities for all
values of
$\kappa>0$. Often (for many values of $\kappa$), when the limsups are
finite
a.s., they are in fact zero, as we show, but the limsups may in some
circumstances take finite, nonzero, values, a.s. In general, the process
crosses one or two-sided boundaries in quite different ways, but
surprisingly
this is not so for the case $\kappa=1/2$. An integral test is given to
distinguish the possibilities in that case. Some results relating to
other
norming sequences for $X$, and when $X$ is centered at a nonstochastic
function, are also given.
http://front.math.ucdavis.edu/math.PR/0603274
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4112. CONSTRUCTIVE NO-ARBITRAGE CRITERION UNDER TRANSACTION COSTS IN
THE CASE OF FINITE DISCRETE TIME
Dmitry B. Rokhlin
We obtain a constructive criterion for robust no-arbitrage in
discrete-time
market models with transaction costs. This criterion is expressed in
terms of
the supports of the regular conditional upper distributions of the
solvency
cones. We also consider the model with a bank account. A method for
construction of arbitrage strategies is proposed.
http://front.math.ucdavis.edu/math.PR/0603284
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4113. INDUCED GELATION IN A TWO-SITE SPATIAL COAGULATION MODEL
Rainer Siegmund-Schultze and Wolfgang Wagner
A two-site spatial coagulation model is considered. Particles of
masses m and
n at the same site form a new particle of mass m+n at rate mn.
Independently,
particles jump to the other site at a constant rate. The limit (for
increasing
particle numbers) of this model is expected to be non-deterministic
after the
gelation time, namely, one or two giant particles randomly jump
between the two
sites. Moreover, a new effect of induced gelation is observed - the
gelation
happening at the site with the larger initial number of monomers
immediately
induces gelation at the other site. Induced gelation is shown to be of
logarithmic order. The limiting behaviour of the model is derived
rigorously up
to the gelation time, while the expected post-gelation behaviour is
illustrated
by a numerical simulation.
http://front.math.ucdavis.edu/math.PR/0603300
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4114. CUBE ROOT FLUCTUATIONS FOR THE CORNER GROWTH MODEL ASSOCIATED
TO THE EXCLUSION PROCESS
Marton Balazs and Eric Cator and Timo Seppalainen
We study the last-passage growth model on the planar integer lattice
with
exponential weights. With boundary conditions that represent the
equilibrium
exclusion process as seen from a particle right after its jump we
prove that
the variance of the last-passage time in a characteristic direction
is of order
t^{2/3}. With more general boundary conditions that include the
rarefaction fan
case we show that the last-passage time fluctuations are still of order
t^{1/3}, and also that the transversal fluctuations of the maximal
path have
order t^{2/3}. We adapt and then build on a recent study of Hammersley's
process by Cator and Groeneboom, and also utilize the competition
interface
introduced by Ferrari, Martin and Pimentel. The arguments are entirely
probabilistic, and no use is made of the combinatorics of Young
tableaux or
methods of asymptotic analysis.
http://front.math.ucdavis.edu/math.PR/0603306
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4115. WEAK DISORDER IN FIBONACCI SEQUENCES
E. Ben-Naim and P.L. Krapivsky
We study how weak disorder affects the growth of the Fibonacci
series. We
introduce a family of stochastic sequences that grow by the normal
Fibonacci
recursion with probability 1-epsilon, but follow a different
recursion rule
with a small probability epsilon. We focus on the weak disorder limit
and
obtain the Lyapunov exponent, that characterizes the typical growth
of the
sequence elements, using perturbation theory. The limiting
distribution for the
ratio of consecutive sequence elements is obtained as well. A number of
variations to the basic Fibonacci recursion including shift,
doubling, and
copying are considered.
http://front.math.ucdavis.edu/cond-mat/0603117
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4116. UNIVERSALITY FOR MATHEMATICAL AND PHYSICAL SYSTEMS
Percy Deift
All physical systems in equilibrium obey the laws of thermodynamics.
In other
words, whatever the precise nature of the interaction between the
atoms and
molecules at the microscopic level, at the macroscopic level,
physical systems
exhibit universal behavior in the sense that they are all governed by
the same
laws and formulae of thermodynamics. In this paper we describe some
recent
history of universality ideas in physics starting with Wigner's model
for the
scattering of neutrons off large nuclei and show how these ideas have
led
mathematicians to investigate universal behavior for a variety of
mathematical
systems. This is true not only for systems which have a physical
origin, but
also for systems which arise in a purely mathematical context such as
the
Riemann hypothesis, and a version of the card game solitaire called
patience
sorting.
http://front.math.ucdavis.edu/math-ph/0603038
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4117. ON FINITE-DIMENSIONAL PROJECTIONS OF DISTRIBUTIONS FOR
SOLUTIONS OF RANDOMLY FORCED PDE'S
Andrei Agrachev (SISSA-Isas) and Sergei Kuksin (Mathematics
Department of Heriot-Watt University), Andrey Sarychev (DMD), Armen
Shirikyan (LM-Orsay)
The paper is devoted to studying the image of probability measures on a
Hilbert space under finite-dimensional analytic maps. We establish
sufficient
conditions under which the image of a measure has a density with
respect to the
Lebesgue measure and continuously depends on the map. The results
obtained are
applied to the 2D Navier--Stokes equations perturbed by various
random forces
of low dimension.
http://front.math.ucdavis.edu/math.AP/0603295
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4118. SIMULATION OF DISCRETE SYSTEMS USING PROBABILISTIC SEQUENTIAL
SYSTEMS
Maria A. Avino-Diaz and Gabriela Bulancea and Oscar Moreno
In this paper we introduce the idea of probability in the definition
of a
Sequential Dynamical System (SDS), thus obtaining a new concept, that of
Probabilistic Sequential System (PSS). Due to its particular dynamic,
the
Probabilistic Boolean Network (PBN) model has been applied to genetic
regulatory networks. The model we introduce combines the sequential
aspect of
the SDSs and the dynamic of the PBNs. The notion of simulation of a
PSS is
introduced using the concept of morphism of PSSs. We prove that the
PSSs with
the PSS-morphisms form a category PSS. Several examples of morphisms,
subsystems and simulations are given.
http://front.math.ucdavis.edu/math.DS/0603289
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4119. SPECIAL HOMOMORPHISMS BETWEEN PROBABILISTIC GENE REGULATORY
NETWORKS
Maria A. Avino-Diaz
In this paper we study finite dynamical systems with $n$ functions
acting on
the same set $X$, and probabilities assigned to these functions, that
it is
called Probabilistic Regulatory Gene Networks (PRN. his concept is
the same or
a natural generalization of the concept Probabilistic Boolean
Networks (PBN),
introduced by I. Shmulevich, E. Dougherty, and W. Zhang, particularly
the model
PBN has been using to describe genetic networks and has therapeutic
applications. In PRNs the most important question is to describe the
steady
states of the systems, so in this paper we pay attention to the idea of
transforming a network to another without lost all the properties, in
particular the probability distribution. Following this objective we
develop
the concepts of homomorphism and $\epsilon$-homomorphism of
probabilistic
regulatory networks, since these concepts bring the properties from one
networks to another. Projections are special homomorphisms, and they
always
induce invariant subnetworks that contain all cycles and steady
states in the
network.
http://front.math.ucdavis.edu/math.DS/0603291
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4120. PROBABILISTIC GENE REGULATORY NETWORKS, ISOMORPHISMS OF MARKOV
CHAINS
Maria A. Avino-Diaz
In this paper we study homomorphisms of Probabilistic Regulatory Gene
Networks(PRN) introduced in arXiv:math.DS/0603289 v1 13 Mar 2006. The
model PRN
is a natural generalization of the Probabilistic Boolean Networks (PBN),
introduced by I. Shmulevich, E. Dougherty, and W. Zhang in 2001, that
has been
using to describe genetic networks and has therapeutic applications.
In this
paper, our main objectives are to apply the concept of homomorphism and
$\epsilon$-homomorphism of probabilistic regulatory networks to the
dynamic of
the networks. The meaning of $\epsilon$ is that these homomorphic
networks have
similar distributions and the distance between the distributions is
upper
bounded by $\epsilon$. Additionally, we prove that the class of PRN
together
with the homomorphisms form a category with products and coproducts.
Projections are special homomorphisms, and they always induce invariant
subnetworks that contain all the cycles and steady states in the
network. Here,
it is proved that the $\epsilon$-homomorphism for $0<\epsilon<1$ produce
simultaneous Markov Chains in both networks, that permit to introduce
the
concept of $\epsilon$-isomorphism of Markov Chains, and similar
networks.
http://front.math.ucdavis.edu/math.DS/0603302
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4121. STATE DEPENDENT UTILITY
Jaime A. Londo\~no
We propose a new approach to utilities that is consistent with
state-dependent utilities. In our model utilities reflect the level of
consumption satisfaction of flows of cash in future times as they are
valued
when the economic agents are making their consumption and investment
decisions.
The theoretical framework used for the model is one proposed by the
author in
Dynamic State Tameness {arXiv:math.PR/0509139}. The proposed
framework is a
generalization of the theory of Brownian flows and can be applied to
those
processes that are the solutions of classical It^o stochastic
differential
equations, even when the volatilities and drifts are just locally
$\delta$-Holder continuous for some $\delta>0$. We develop the
martingale
methodology for the solution of the problem of optimal consumption and
investment. Complete solutions of the optimal consumption and
portfolio problem
are obtained in a very general setting which includes several
functional forms
for utilities in the current literature, and consider general
restrictions on
minimal wealths. As a secondary result we obtain a suitable
representation for
straightforward numerical computations of the optimal consumption and
investment strategies.
http://front.math.ucdavis.edu/math.PR/0603316
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4122. SYSTEMATIC SCAN FOR SAMPLING COLORINGS
Martin Dyer and Leslie Ann Goldberg and Mark Jerrum
We address the problem of sampling colorings of a graph $G$ by Markov
chain
simulation. For most of the article we restrict attention to proper
$q$-colorings of a path on $n$ vertices (in statistical physics
terms, the
one-dimensional $q$-state Potts model at zero temperature), though in
later
sections we widen our scope to general ``$H$-colorings'' of arbitrary
graphs
$G$. Existing theoretical analyses of the mixing time of such
simulations
relate mainly to a dynamics in which a random vertex is selected for
updating
at each step. However, experimental work is often carried out using
systematic
strategies that cycle through coordinates in a deterministic manner,
a dynamics
sometimes known as systematic scan. The mixing time of systematic
scan seems
more difficult to analyze than that of random updates, and little is
currently
known. In this article we go some way toward correcting this
imbalance. By
adapting a variety of techniques, we derive upper and lower bounds
(often
tight) on the mixing time of systematic scan. An unusual feature of
systematic
scan as far as the analysis is concerned is that it fails to be time
reversible.
http://front.math.ucdavis.edu/math.PR/0603323
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4123. PATTERN DENSITIES IN FLUID DIMER MODELS
Cedric Boutillier
In this paper, we introduce a family of observables for the dimer
model on a
bi-periodic bipartite planar graph, called pattern density fields. We
study the
scaling limit of these objects for liquid and gaseous Gibbs measures
of the
dimer model, and prove that they converge to a linear combination of a
derivative of the Gaussian massless free field and an independent
white noise.
http://front.math.ucdavis.edu/math.PR/0603324
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4124. MEAN FIELD CONVERGENCE OF A MODEL OF MULTIPLE TCP CONNECTIONS
THROUGH A BUFFER IMPLEMENTING RED
D. R. McDonald and J. Reynier
RED (Random Early Detection) has been suggested when multiple TCP
sessions
are multiplexed through a bottleneck buffer. The idea is to detect
congestion
before the buffer overflows by dropping or marking packets with a
probability
that increases with the queue length. The objectives are reduced
packet loss,
higher throughput, reduced delay and reduced delay variation achieved
through
an equitable distribution of packet loss and reduced synchronization.
Baccelli,
McDonald and Reynier [Performance Evaluation 11 (2002) 77--97] have
proposed a
fluid model for multiple TCP connections in the congestion avoidance
regime
multiplexed through a bottleneck buffer implementing RED. The window
sizes of
each TCP session evolve like independent dynamical systems coupled by
the queue
length at the buffer. The key idea in [Performance Evaluation 11
(2002) 77--97]
is to consider the histogram of window sizes as a random measure
coupled with
the queue. Here we prove the conjecture made in [Performance
Evaluation 11
(2002) 77--97] that, as the number of connections tends to infinity,
this
system converges to a deterministic mean-field limit comprising the
window size
density coupled with a deterministic queue.
http://front.math.ucdavis.edu/math.PR/0603325
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4125. LARGE DEVIATION ASYMPTOTICS AND CONTROL VARIATES FOR SIMULATING
LARGE FUNCTIONS
Sean P. Meyn
Consider the normalized partial sums of a real-valued function $F$ of a
Markov chain, \[\phi_n:=n^{-1}\sum_{k=0}^{n-1}F(\Phi(k)),\qquad n\ge1.
\] The
chain $\{\Phi(k):k\ge0\}$ takes values in a general state space $
\mathsf {X}$,
with transition kernel $P$, and it is assumed that the Lyapunov drift
condition
holds: $PV\le V-W+b\mathbb{I}_C$ where $V:\mathsf {X}\to(0,\infty)$,
$W:\mathsf
{X}\to[1,\infty)$, the set $C$ is small and $W$ dominates $F$. Under
these
assumptions, the following conclusions are obtained: 1. It is known
that this
drift condition is equivalent to the existence of a unique invariant
distribution $\pi$ satisfying $\pi(W)<\infty$, and the law of large
numbers
holds for any function $F$ dominated by $W$:
\[\phi_n\to\phi:=\pi(F),\qquad{a.s.}, n\to\infty.\] 2. The lower error
probability defined by $\mathsf {P}\{\phi_n\le c\}$, for $c<\phi$, $n
\ge1$,
satisfies a large deviation limit theorem when the function $F$
satisfies a
monotonicity condition. Under additional minor conditions an exact large
deviations expansion is obtained. 3. If $W$ is near-monotone, then
control-variates are constructed based on the Lyapunov function $V$,
providing
a pair of estimators that together satisfy nontrivial large
asymptotics for the
lower and upper error probabilities. In an application to simulation
of queues
it is shown that exact large deviation asymptotics are possible even
when the
estimator does not satisfy a central limit theorem.
http://front.math.ucdavis.edu/math.PR/0603328
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4126. CORRECTION. IMPROPER REGULAR CONDITIONAL DISTRIBUTIONS
Teddy Seidenfeld and Mark J. Schervish and Joseph B. Kadane
Correction to Annals of Probability 29 (2001) 1612--1624
[doi:10.1214/aop/1015345764].
http://front.math.ucdavis.edu/math.PR/0603012
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4127. ASYMPTOTIC THEOREMS OF SEQUENTIAL ESTIMATION-ADJUSTED URN MODELS
Li-X. Zhang and Feifang Hu and Siu Hung Cheung
The Generalized P\'{o}lya Urn (GPU) is a popular urn model which is
widely
used in many disciplines. In particular, it is extensively used in
treatment
allocation schemes in clinical trials. In this paper, we propose a
sequential
estimation-adjusted urn model (a nonhomogeneous GPU) which has a wide
spectrum
of applications. Because the proposed urn model depends on sequential
estimations of unknown parameters, the derivation of asymptotic
properties is
mathematically intricate and the corresponding results are
unavailable in the
literature. We overcome these hurdles and establish the strong
consistency and
asymptotic normality for both the patient allocation and the
estimators of
unknown parameters, under some widely satisfied conditions. These
properties
are important for statistical inferences and they are also useful for
the
understanding of the urn limiting process. A superior feature of our
proposed
model is its capability to yield limiting treatment proportions
according to
any desired allocation target. The applicability of our model is
illustrated
with a number of examples.
http://front.math.ucdavis.edu/math.PR/0603329
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4128. ON THE ASYMPTOTICS OF THE SUPREMUM OF A RANDOM WALK: THE
PRINCIPLE OF A SINGLE BIG JUMP IN THE LIGHT-TAILED CASE
Stan Zachary and Serguei Foss
We study the distribution of the maximum $M$ of a random walk whose
increments have a distribution with negative mean and belonging, for
some
$\gamma\ge0$, to the class $\mathcal{S}_{\gamma}$ introduced by
Chover, Ney,
and Weinger (1973). For $\gamma>0$, we give a probabilistic
derivation of the
asymptotic tail distribution of $M$ and show that, as in the case $
\gamma=0$,
extreme values of $M$ are in general attained through some single large
increment in the random walk.
http://front.math.ucdavis.edu/math.PR/0603330
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4129. INDIVIDUAL VERSUS CLUSTER RECOVERIES WITHIN A SPATIALLY
STRUCTURED POPULATION
L. Belhadji and N. Lanchier
Stochastic modeling of disease dynamics has had a long tradition.
Among the
first epidemic models including a spatial structure in the form of local
interactions is the contact process. In this article we investigate two
extensions of the contact process describing the course of a single
disease
within a spatially structured human population distributed in social
clusters.
That is, each site of the $d$-dimensional integer lattice is occupied
by a
cluster of individuals; each individual can be healthy or infected. The
evolution of the disease depends on three parameters, namely the outside
infection rate which models the interactions between the clusters,
the within
infection rate which takes into account the repeated contacts between
individuals in the same cluster, and the size of each social cluster.
For the
first model, we assume cluster recoveries, while individual
recoveries are
assumed for the second one. The aim is to investigate the existence of
nontrivial stationary distributions for both processes depending on
the value
of each of the three parameters. Our results show that the
probability of an
epidemic strongly depends on the recovery mechanism.
http://front.math.ucdavis.edu/math.PR/0603331
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4130. A ZERO-ONE LAW FOR FIRST-ORDER LOGIC ON RANDOM IMAGES
David Coupier (MAP5) and Agn\`{e}s Desolneux (MAP5) and Bernard
Ycart (LMC - IMAG)
For an $n\times n$ random image with independent pixels, black with
probability $p(n)$ and white with probability $1-p(n)$, the
probability of
satisfying any given first-order sentence tends to 0 or 1, provided both
$p(n)n^{\frac{2}{k}}$ and $(1-p(n))n^{\frac{2}{k}}$ tend to 0 or $+
\infty$, for
any integer $k$. The result is proved by computing the threshold
function for
basic local sentences, and applying Gaifman's theorem.
http://front.math.ucdavis.edu/math.PR/0603333
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4131. SOME STRONG LIMIT THEOREMS FOR THE LARGEST ENTRIES OF SAMPLE
CORRELATION MATRICES
Deli Li and Andrew Rosalsky
Let $\{X_{k,i};i\geq 1,k\geq 1\}$ be an array of i.i.d. random
variables and
let $\{p_n;n\geq 1\}$ be a sequence of positive integers such that $n/
p_n$ is
bounded away from 0 and $\infty$. For $W_n=\max_{1\leq i<j\leq
p_n}|\sum_{k=1}^nX_{k,i}X_{k,j}|$ and $L_n=\max_{1\leq i<j\leq
p_n}|\hat{\rho}^{(n)}_{i,j}|$ where $\hat{\rho}^{(n)}_{i,j}$ denotes the
Pearson correlation coefficient between $(X_{1,i},...,X_{n,i})'$ and
$(X_{1,j},...,X_{n,j})'$, the limit laws (i) $\lim_{n\to
\infty}\frac{W_n}{n^{\alpha}}=0$ a.s. $(\alpha >1/2)$, (ii) $\lim_{n\to
\infty}n^{1-\alpha}L_n=0$ a.s. $(1/2<\alpha \leq 1)$, (iii) $\lim_{n\to
\infty}\frac{W_n}{\sqrt{n\log n}}=2$ a.s. and (iv) $\lim_{n\to
\infty}(\frac{n}{\log n})^{1/2}L_n=2$ a.s. are shown to hold under
optimal sets
of conditions. These results follow from some general theorems proved
for
arrays of i.i.d. two-dimensional random vectors. The converses of the
limit
laws (i) and (iii) are also established. The current work was
inspired by
Jiang's study of the asymptotic behavior of the largest entries of
sample
correlation matrices.
http://front.math.ucdavis.edu/math.PR/0603334
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4132. STOCHASTIC SPATIAL MODELS OF HOST-PATHOGEN AND HOST-MUTUALIST
INTERACTIONS I
N. Lanchier and C. Neuhauser
Mutualists and pathogens, collectively called symbionts, are
ubiquitous in
plant communities. While some symbionts are highly host-specific, others
associate with multiple hosts. The outcomes of multispecies host-
symbiont
interactions with different degrees of specificity are difficult to
predict at
this point due to a lack of a general conceptual framework.
Complicating our
predictive power is the fact that plant populations are spatially
explicit, and
we know from past research that explicit space can profoundly alter
plant-plant
interactions. We introduce a spatially explicit, stochastic model to
investigate the role of explicit space and host-specificity in
multispecies
host-symbiont interactions. We find that in our model, pathogens can
significantly alter the spatial structure of plant communities,
promoting
coexistence, whereas mutualists appear to have only a limited effect.
Effects
are more pronounced the more host-specific symbionts are.
http://front.math.ucdavis.edu/math.PR/0603335
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4133. IMAGE DENOISING BY STATISTICAL AREA THRESHOLDING
David Coupier (MAP5) and Agn\`{e}s Desolneux (MAP5) and Bernard
Ycart (LMC - IMAG)
Area openings and closings are morphological filters which efficiently
suppress impulse noise from an image, by removing small connected
components of
level sets. The problem of an objective choice of threshold for the area
remains open. Here, a mathematical model for random images will be
considered.
Under this model, a Poisson approximation for the probability of
appearance of
any local pattern can be computed. In particular, the probability of
observing
a component with size larger than $k$ in pure impulse noise has an
explicit
form. This permits the definition of a statistical test on the
significance of
connected components, thus providing an explicit formula for the area
threshold
of the denoising filter, as a function of the impulse noise probability
parameter. Finally, using threshold decomposition, a denoising
algorithm for
grey level images is proposed.
http://front.math.ucdavis.edu/math.PR/0603337
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4134. THE ARCSINE LAW AS A UNIVERSAL AGING SCHEME FOR TRAP MODELS
Gerard Ben Arous and Jiri Cerny
We give a general proof of aging for trap models using the arcsine
law for
stable subordinators. This proof is based on abstract conditions on the
potential theory of the underlying graph and on the randomness of the
trapping
landscape. We apply this proof to aging for trap models on large
two-dimensional tori and for trap dynamics of the Random Energy Model
on a
broad range of time scales.
http://front.math.ucdavis.edu/math.PR/0603340
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4135. DISCRETE IT\^O FORMULAS AND THEIR APPLICATIONS TO STOCHASTIC
NUMERICS
Jir\^o Akahori
This is a survey note of the author's observations on the discrete-time
analogues of It\^o formulas.
http://front.math.ucdavis.edu/math.PR/0603341
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4136. DYNAMICS OF TRAP MODELS
Gerard Ben Arous and Jiri Cerny
These notes cover one of the topics of the class given in the Les
Houches
Summer School ``Mathematical statistical physics'' in July 2005. The
lectures
tried to give a summary of the recent mathematical results about the
long-time
behaviour of dynamics of (mean-field) spin-glasses and other
disordered media.
We have chosen here to restrict the scope of these notes to the
dynamics of
trap models only, but to cover this topic in somewhat more depth.
http://front.math.ucdavis.edu/math.PR/0603344
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4137. CORRECTION. CENTRAL LIMIT THEOREMS FOR ADDITIVE FUNCTIONALS OF
THE SIMPLE EXCLUSION PROCESS
S. Sethuraman
Correction to Annals of Probability 28 (2000) 277--302
[doi:10.1214/aop/1019160120].
http://front.math.ucdavis.edu/math.PR/0603014
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4138. SECOND CLASS PARTICLES AND CUBE ROOT ASYMPTOTICS FOR
HAMMERSLEY'S PROCESS
Eric Cator and Piet Groeneboom
We show that, for a stationary version of Hammersley's process, with
Poisson
sources on the positive x-axis and Poisson sinks on the positive y-
axis, the
variance of the length of a longest weakly North-East path $L(t,t)$ from
$(0,0)$ to $(t,t)$ is equal to $2\E(t-X(t))_+$, where $X(t)$ is the
location of
a second class particle at time $t$. This implies that both $\E(t-X
(t))_+$ and
the variance of $L(t,t)$ are of order $t^{2/3}$. Proofs are based on the
relation between the flux and the path of a second class particle,
continuing
the approach of Cator and Groeneboom (2005).
http://front.math.ucdavis.edu/math.PR/0603345
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4139. RIGHT-PERMUTATIVE CELLULAR AUTOMATA ON TOPOLOGICAL MARKOV CHAINS
Marcelo Sobottka
In this paper we consider cellular automata $(\mathfrak{G},\Phi)$ with
algebraic local rules and such that $\mathfrak{G}$ is a topological
Markov
chain which has a structure compatible to this local rule. We
characterize such
cellular automata and study the convergence of the Ces\`aro mean
distribution
of the iterates of any probability measure with complete connections and
summable decay.
http://front.math.ucdavis.edu/math.DS/0603326
---------------------------------------------------------------
4140. A SUBDIFFUSIVE BEHAVIOUR OF RECURRENT RANDOM WALK IN RANDOM
ENVIRONMENT ON A REGULAR TREE
Yueyun Hu (LAGA) and Zhan Shi (PMA)
We are interested in the random walk in random environment on an
infinite
tree. Lyons and Pemantle [11] give a precise recurrence/transience
criterion.
Our paper focuses on the almost sure asymptotic behaviours of a
recurrent
random walk $(X\_n)$ in random environment on a regular tree, which
is closely
related to Mandelbrot [13]'s multiplicative cascade. We prove, under
some
general assumptions upon the distribution of the environment, the
existence of
a new exponent $\nu\in (0, {1\over 2}]$ such that $\max\_{0\le i \le
n} |X\_i|$
behaves asymptotically like $n^{\nu}$. The value of $\nu$ is explicitly
formulated in terms of the distribution of the environment.
http://front.math.ucdavis.edu/math.PR/0603363
---------------------------------------------------------------
4141. BETA-PATHS IN THE HAMMERSLEY PROCESS
Cristian Coletti and Leandro P. R. Pimentel
We study the asymptotics of beta-paths in the Hammersley process with
sources
and sinks, in the rarefaction regime. We derive a strong law of large
number
for those paths and we show that its fluctuation exponent is at most
2/3.
Examples of beta-paths are the space-time path of a second-class
particle in
the Hammersley process and also the space-time path of the interface
between
two PNG droplets.
http://front.math.ucdavis.edu/math.PR/0603382
---------------------------------------------------------------
4142. TESTING STATISTICAL HYPOTHESIS ON RANDOM TREES
Jorge R. Busch and Pablo A. Ferrari and A. Georgina Flesia and
Ricardo Fraiman and Sebastian Grynberg
To distinguish between populations of trees, we consider the
hypothesis test
proposed recently by Balding, Ferrari, Fraiman and Sued (BFFS--test).
A direct
approach to calculate effectively the test statistic is quite
difficult, since
it is based on a supremum defined over the space of all trees, which
grows
exponentially fast. We show how to transform this problem into a max-
flow over
a network which can be solved using a Ford Fulkerson algorithm in
polynomial
time on the maximal number of vertices of the random tree. We also
describe
conditions that imply the characterization of the measure by the
marginal
distributions of each node of the random tree, which validate the use
of the
BFFS--test for measure discrimination. The performance of the test is
studied
via simulations on Galton-Watson processes.
http://front.math.ucdavis.edu/math.ST/0603378
---------------------------------------------------------------
4143. DEVIATION BOUNDS FOR ADDITIVE FUNCTIONALS OF MARKOV PROCESS
Patrick Cattiaux (CMAP and Modal'x) and Arnaud Guillin (CEREMADE)
In this paper we derive non asymptotic deviation bounds for $$\P_\nu
(|\frac
1t \int_0^t V(X_s) ds - \int V d\mu | \geq
R)$$ where $X$ is a $\mu$ stationary and ergodic Markov process
and $V$ is
some $\mu$ integrable function. These bounds are obtained under
various moments
assumptions for $V$, and various regularity assumptions for $\mu$.
Regularity
means here that $\mu$ may satisfy various functional inequalities (F-
Sobolev,
generalized Poincar\'e etc...).
http://front.math.ucdavis.edu/math.PR/0603021
---------------------------------------------------------------
4144. WEAK DISORDER FOR LOW DIMENSIONAL POLYMERS: THE MODEL OF STABLE
LAWS
Francis Comets (PMA)
In this paper, we consider directed polymers in random environment
with long
range jumps in discrete space and time. We extend to this case some
techniques,
results and classifications known in the usual short range case.
However, some
properties are drastically different when the underlying random walk
belongs to
the domain of attraction of an $\a$-stable law. For instance, we
construct
natural examples of directed polymers in random environment which
experience
weak disorder in low dimension.
http://front.math.ucdavis.edu/math.PR/0603390
---------------------------------------------------------------
4145. TRANSIENT RANDOM WALKS ON A STRIP IN A RANDOM ENVIRONMENT
Alexander Roitershtein
We consider transient random walks on a strip in a random
environment. The
model was introduced by Bolthausen and Goldsheid in [4]. We derive a
strong law
of large numbers for the random walks in a general ergodic setup and
obtain an
annealed central limit theorem in the case of uniformly mixing
environments. In
addition, we prove that the law of the ``environment viewed from the
position
of the walker'' converges to a limiting distribution if the
environment is an
i.i.d. sequence.
http://front.math.ucdavis.edu/math.PR/0603392
---------------------------------------------------------------
4146. PROCESS LEVEL MODERATE DEVIATIONS FOR STABILIZING FUNCTIONALS
Peter Eichelsbacher and Tomasz Schreiber
Functionals of spatial point process often satisfy a weak spatial
dependence
condition known as stabilization. In this paper we prove process
level moderate
deviation principles (MDP) for such functionals, which are a level-3
result for
empirical point fields as well as a level-2 result for empirical point
measures. The level-3 rate function coincides with the so-called
specific
information. We show that the general result can be applied to prove
MDPs for
various particular functionals, including random sequential packing,
birth-growth models, germ-grain models and nearest neighbor graphs.
http://front.math.ucdavis.edu/math.PR/0603402
---------------------------------------------------------------
4147. SOME SCALING LIMITS FOR A BROWNIAN POLYMER IN A GAUSSIAN MEDIUM
Sergio De Carvalho Bezerra (IECN) and Samy Tindel (IECN) and
Frederi Viens
This paper provides information about the asymptotic behavior of a
one-dimensional Brownian polymer in random medium represented by a
space-time
Gaussian field W assumed to be white noise in time and function-
valued in
space. According to the behavior of the spatial covariance W, we give
sharp
upper and lower bounds on the partition function's exponential rate
(Lyapunov
exponent), and on the growth (wandering exponent) of the polymer
itself when
the time parameter goes to infinity.
http://front.math.ucdavis.edu/math.PR/0603404
---------------------------------------------------------------
4148. LARGE DEVIATIONS FOR PAST-DEPENDENT RECURSIONS
F. Klebaner and R. Liptser
The Large Deviation Principle is established for stochastic models
defined by
past-dependent non linear recursions with small noise. In the Markov
case we
use the result to obtain an explicit expression for the asymptotics
of exit
time.
http://front.math.ucdavis.edu/math.PR/0603407
---------------------------------------------------------------
4149. PERMUTATIONS WITHOUT LONG DECREASING SUBSEQUENCES AND RANDOM
MATRICES
Piotr Sniady
We study the shape of the Young diagram \lambda associated via the
Robinson-Schensted-Knuth algorithm to a random permutation in S_n
such that the
length of the longest decreasing subsequence is not bigger than a
fixed number
d; in other words we study the restriction of the Plancherel measure
to Young
diagrams with at most d rows. We prove that in the limit n\to\infty
the rows of
\lambda behave like the eigenvalues of a certain random matrix
(traceless
Gaussian Unitary Ensemble) with d rows and columns. In particular,
the length
of the longest increasing subsequence of such a random permutation
behaves
asymptotically like the largest eigenvalue of the corresponding
random matrix.
http://front.math.ucdavis.edu/math.CO/0603401
---------------------------------------------------------------
4150. WEAK SURVIVAL FOR BRANCHING RANDOM WALKS ON GRAPHS
Daniela Bertacchi and Fabio Zucca
We study weak and strong survival for branching random walks on
multigraphs.
We prove that, for a large class of multigraphs, weak survival is
related to a
geometrical parameter of the multigraph and that the existence of a
pure weak
phase is equivalent to nonamenability. Finally we study weak and strong
critical behaviors of the branching random walk.
http://front.math.ucdavis.edu/math.PR/0603412
---------------------------------------------------------------
4151. MODERATE DEVIATIONS FOR SOME POINT MEASURES IN GEOMETRIC
PROBABILITY
Peter Eichelsbacher and Tomasz Schreiber and Joseph E. Yukich
Functionals in geometric probability are often expressed as sums of
bounded
functions exhibiting exponential stabilization.
Methods based on cumulant techniques and exponential modifications
of measures
show that such functionals satisfy moderate deviation principles.
This leads to
moderate deviation principles and laws of the iterated logarithm for
random
packing models as well as for statistics associated with germ-grain
models and
$k$ nearest neighbor graphs.
http://front.math.ucdavis.edu/math.PR/0603022
---------------------------------------------------------------
4152. CONVEX GEOMETRY OF MAX-STABLE DISTRIBUTIONS
Ilya Molchanov
It is shown that max-stable random vectors in $[0,\infty)^d$ with unit
Fr\'echet marginals are in one to one correspondence with convex sets
$K$ in
$[0,\infty)^d$ called max-zonoids. The max-zonoids can be
characterised as sets
obtained as limits of Minkowski sums of simplices or, alternatively,
as the
selection expectation of a random simplex whose distribution is
controlled by
the spectral measure of the max-stable random vector. Furthermore, the
cumulative distribution function $\Prob{\xi\leq x}$ of a max-stable
random
vector $\xi$ with unit Fr\'echet marginals is determined by the norm
of the
inverse to $x$, where all possible norms are given by the support
functions of
max-zonoids. As an application, geometrical interpretations of a
number of
well-known concepts from the theory of multivariate extreme values
and copulas
are provided. The convex geometry approach makes it possible to
generalise a
number of known results and to introduce new operations with max-
stable random
vectors.
http://front.math.ucdavis.edu/math.PR/0603423
---------------------------------------------------------------
4153. DIFFERENTIABILITY OF BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS
IN HILBERT SPACES WITH MONOTONE GENERATORS
Philippe Briand (IRMAR) and Fulvia Confortola
The aim of the present paper is to study the regularity properties of
the
solution of a backward stochastic differential equation with a monotone
generator in infinite dimension. We show some applications to the
nonlinear
Kolmogorov equation and to stochastic optimal control.
http://front.math.ucdavis.edu/math.PR/0603428
---------------------------------------------------------------
4154. ON THE INFERENCE OF SPARTAN SPATIAL RANDOM FIELD MODELS FOR
GEOSTATISTICAL APPLICATIONS
Samuel Elogne and Dionisis Hristopulos
This paper focuses on the estimation of model parameters (model
inference)
for the class of Spartan Spatial Random Fields (SSRFs) introduced by
Hristopulos (2003). The approach used for model inference involves
calculation
of sample constraints and fitting with respective ensemble
constraints. The
fitting leads to optimal SSRF parameters obtained by minimizing a
suitable
distance functional. We propose kernel-based estimators for
calculating the
sample constraints from data distributed on irregular sampling grids. We
investigate the asymptotic properties of the estimators, and we
establish a
criterion for the selection of the kernel bandwidth parameters. The
performance
of the sample constraint estimators, as well as that of the SSRF
inference
procedure is evaluated by means of numerical simulations for
different models
of spatial dependence.
http://front.math.ucdavis.edu/math.ST/0603430
---------------------------------------------------------------
4155. CONVERGENCE OF APPROXIMATIONS OF MONOTONE GRADIENT SYSTEMS
Lorenzo Zambotti
We consider stochastic differential equations in a Hilbert space,
perturbed
by the gradient of a convex potential. We investigate the problem of
convergence of a sequence of such processes. We propose applications
of this
method to reflecting O.U. processes in infinite dimension, to stochastic
partial differential equations with reflection of Cahn-Hilliard type
and to
interface models.
http://front.math.ucdavis.edu/math.PR/0603474
---------------------------------------------------------------
4156. STATISTICAL PROPERTIES OF TOPOLOGICAL COLLET-ECKMANN MAPS
Feliks Przytycki and Juan Rivera-Letelier
We study geometric and statistical properties of complex rational maps
satisfying the Topological Collet-Eckmann Condition. We show that
every such a
rational map possesses a unique conformal probability measure of minimal
exponent, and that this measure is non-atomic, ergodic and that its
Hausdorff
dimension is equal to the Hausdorff dimension of the Julia set.
Furthermore, we
show that there is a unique invariant probability measure that is
absolutely
continuous with respect to this conformal measure, and we show that this
measure is exponentially mixing (it has exponential decay of
correlations) and
that it satisfies the Central Limit Theorem.
We also show that for a complex rational map f the existence of
such an
invariant measure characterizes the Topological Collet-Eckmann
Condition, and
that this measure is the unique equilibrium state with potential - HD
(J(f)) ln
|f'|.
http://front.math.ucdavis.edu/math.DS/0603459
---------------------------------------------------------------
4157. ON INEQUALITIES FOR SUMS OF BOUNDED RANDOM VARIABLES
Iosif Pinelis
Let $\eta_1,\eta_2,...$ be independent (but not necessarily identically
distributed) zero-mean random variables (r.v.'s) such that $|\eta_i|
\le1$
almost surely for all $i$, and let $Z$ stand for a standard normal
r.v. Let
$a_1,a_2,...$ be any real numbers such that $a_1^2+a_2^2+...=1.$ It
is shown
that then for all $x>0$ $$ \P(a_1\eta_1+a_2\eta_2+...\ge x) \le \P(Z\ge
x-\la/x), $$ where $\la := \ln\frac{2e^3}9=1.495...$. The proof
relies on (i)
another probability inequality and (ii) a l'Hospital-type rule for
monotonicity, both developed elsewhere. Extensions to (super)
martingales are
indicated.
http://front.math.ucdavis.edu/math.PR/0603030
---------------------------------------------------------------
4158. THEOREMS LIMIT WITH WEIGHT FOR THE VECTORIAL MARTINGALES TO
CONTINUOUS TIME
Faouzi Chaabane and Ahmed Kebaier
We develop a general approach of the almost sure central limit
theorem for
the quasi-continuous vectorial martingales and we release a quadratic
extension
of this theorem while specifying speeds of convergence. As an
application of
this result we study the problem of estimate the variance of a
process with
stationary and idependent increments in statistics.
http://front.math.ucdavis.edu/math.PR/0603492
---------------------------------------------------------------
4159. EXPLICIT LAWS OF LARGE NUMBERS FOR RANDOM NEAREST-NEIGHBOUR
TYPE GRAPHS
Andrew R. Wade
We give laws of large numbers (in the L^p sense) for the total length
of the
k-nearest neighbours (directed) graph and the j-th nearest neighbour
(directed)
graph in R^d, with power-weighted edges. We deduce a law of large
numbers for
the standard nearest neighbour (undirected) graph. We give the limiting
constants, in the case of uniform random points in (0,1)^d,
explicitly. Also,
we give explicit laws of large numbers for the total power-weighted
length of
the Gabriel graph and two further graphs that are related to the
standard
nearest-neighbour graph: the on-line nearest-neighbour graph and the
minimal
directed spanning forest.
http://front.math.ucdavis.edu/math.PR/0603559
---------------------------------------------------------------
4160. LIMIT THEORY FOR THE RANDOM ON-LINE NEAREST-NEIGHBOUR GRAPH
Mathew D. Penrose and Andrew R. Wade
In the on-line nearest-neighbour graph (ONG), each point after the
first in a
sequence of points in R^d is joined by an edge to its nearest-
neighbour amongst
those points that precede it in the sequence. We study the large-sample
asymptotic behaviour of the total power-weighted length of the ONG on
uniform
random points in (0,1)^d. In particular, for d=1 and weight exponent
\alpha>1/2, the limiting distribution of the centred total weight is
characterized by a distributional fixed-point equation. As an
ancillary result,
we give exact expressions for the expectation and variance of the
standard
nearest-neighbour (directed) graph on uniform random points in the unit
interval.
http://front.math.ucdavis.edu/math.PR/0603561
---------------------------------------------------------------
4161. A GENERALIZATION OF THE CENTRAL LIMIT THEOREM CONSISTENT WITH
NONEXTENSIVE STATISTICAL MECHANICS
Sabir Umarov and Stanly Steinberg and Constantino Tsallis
As well known, the standard central limit theorem plays a fundamental
role in
Boltzmann-Gibbs (BG) statistical mechanics. This important physical
theory has
been generalized by one of us (CT) in 1988 by using the entropy $S_q =
\frac{1-\sum_i p_i^q}{q-1}$ (with $q \in \cal{R}$) instead of its
particular
case $S_1=S_{BG}= -\sum_i p_i \ln p_i$. The theory which emerges is
usually
referred to as {\it nonextensive statistical mechanics} and recovers the
standard theory for $q=1$. During the last two decades, this $q$-
generalized
statistical mechanics has been successfully applied to a considerable
amount of
physically interesting complex phenomena. Conjectures and numerical
indications
available in the literature were since a few years suggesting the
possibility
of $q$-generalizations of the standard central limit theorem by
allowing the
random variables that are being summed to be correlated in some
special manner,
the case $q=1$ corresponding to standard probabilistic independence.
This is
precisely what we prove in the present paper for some range of $q$ which
extends from below to above $q=1$. The attractor, in the usual sense
of a
central limit theorem, is given by a distribution of the form $p(x)
\propto
[1-(1-q) \beta x^2]^{1/(1-q)}$ with $\beta>0$. These distributions,
sometimes
referred to as $q$-Gaussians, are known to make, under appropriate
constraints,
extremal the functional $S_q$. Their $q=1$ and $q=2$ particular cases
recover
respectively Gaussian and Cauchy distributions.
http://front.math.ucdavis.edu/cond-mat/0603593
---------------------------------------------------------------
4162. A GEOMETRICAL STRUCTURE FOR AN INFINITE ORIENTED CLUSTER AND
ITS UNIQUENESS
Xian-Yuan Wu and Yu Zhang
We consider the supercritical oriented percolation model. Let ${\fK}$
be all
the percolation points. For each $u\in {\fK}$, we write $\gamma_u$ as
its
right-most path. Let $G=\cup_u \gamma_u$. In this paper, we show that
$G$ is a single tree with only one topological end. We also present a
relationship between ${\fK}$ and $G$ and construct a bijection
between ${\fK}$
and $\Z$ using the preorder traversal algorithm. Through applications
of this
fundamental graph property, we show the uniqueness of an infinite
oriented
cluster by ignoring finite vertices.
http://front.math.ucdavis.edu/math.PR/0603580
---------------------------------------------------------------
4163. ULTRAMETRIC RANDOM FIELD
A.Yu.Khrennikov and S.V.Kozyrev
Gaussian random field on general ultrametric space is introduced as a
solution of pseudodifferential stochastic equation. Covariation of the
introduced random field is computed with the help of wavelet analysis on
ultrametric spaces.
Notion of ultrametric Markovianity, which describes independence of
contributions to random field from different ultrametric balls is
introduced.
We show that the random field under investigation satisfies this
property.
http://front.math.ucdavis.edu/math.PR/0603584
---------------------------------------------------------------
4164. ON RAW CODING OF CHAOTIC DYNAMICS
Michael Blank
We study raw coding of trajectories of a chaotic dynamical system by
sequences of elements from a finite alphabet and show that there is a
fundamental constraint on differences between codes corresponding to
different
trajectories of the dynamical system.
http://front.math.ucdavis.edu/math.DS/0603575
---------------------------------------------------------------
4165. CONDITIONED STABLE L\'{E}VY PROCESSES AND LAMPERTI REPRESENTATION
Maria Emilia Caballero and Lo\"{i}c Chaumont (PMA)
By killing a stable L\'{e}vy process when it leaves the positive half-
line,
or by conditioning it to stay positive, or by conditioning it to hit 0
continuously, we obtain three different positive self-similar Markov
processes
which illustrate the three classes described by Lamperti \cite{La}.
For each of
these processes, we compute explicitly the infinitesimal generator
from which
we deduce the characteristics of the underlying L\'{e}vy process in the
Lamperti representation. The proof of this result bears on the
behaviour at
time 0 of stable L\'{e}vy processes before their first passage time
across
level 0 which we describe here. As an application, we give the law of
the
minimum before an independent exponential time of a certain class of L
\'{e}vy
processes. It provides the explicit form of the spacial Wiener-Hopf
factor at a
particular point and the value of the ruin probability for this class of
L\'{e}vy processes.
http://front.math.ucdavis.edu/math.PR/0603613
---------------------------------------------------------------
4166. SCATTERING LENGTH FOR STABLE PROCESSES
B. Siudeja
Let $\alpha\in(0,2)$ and $X_t$ be a symmetric $\alpha$-stable
process. We
define the scattering length $\Gamma(v)$ of the positive potential $v
$ and
prove several of its basic properties. We use the scattering length to
findestimates for the first eigenvalue of the Schr\"odinger operator
of the
``Neumann'' fractional Laplacian in a cube with potential $v$.
http://front.math.ucdavis.edu/math.PR/0603627
---------------------------------------------------------------
4167. ON THE NUMBER OF CIRCUITS IN RANDOM GRAPHS
Enzo Marinari and Guilhem Semerjian
We apply in this article (non rigorous) statistical mechanics methods
to the
problem of counting long circuits in graphs. The outcomes of this
approach have
two complementary flavours. On the algorithmic side, we propose an
approximate
counting procedure, valid in principle for a large class of graphs.
On a more
theoretical side, we study the typical number of long circuits in
random graph
ensembles, reproducing rigorously known results and stating new
conjectures.
http://front.math.ucdavis.edu/cond-mat/0603657
---------------------------------------------------------------
4168. EXISTENCE OF SADDLE POINTS IN DISCRETE MARKOV GAMES AND ITS
APPLICATION IN NUMERICAL METHODS FOR STOCHASTIC DIFFERENTIAL GAMES
Q. S. Song and G. Yin
This work establishes sufficient conditions for existence of saddle
points in
discrete Markov games. The result reveals the relation between
dynamic games
and static games using dynamic programming equations. This result
enables us to
prove existence of saddle points of non-separable stochastic
differential games
of regime-switching diffusions under appropriate conditions.
http://front.math.ucdavis.edu/math.OC/0603600
---------------------------------------------------------------
4169. BESSEL CONVOLUTIONS ON MATRIX CONES: ALGEBRAIC PROPERTIES AND
RANDOM WALKS
Michael Voit
Bessel-type convolution algebras of bounded Borel measures on the matrix
cones of positive semidefinite $q\times q$-matrices over $\mathbb R,
\mathbb C,
\mathbb H$ were introduced recently by R\"osler. These convolutions
depend on
some continuous parameter, generate commutative hypergroup structures
and have
Bessel functions of matrix argument as characters.
Here, we first study the rich algebraic structure of these
hypergroups. In
particular, the subhypergroups and automorphisms are classified, and
we show
that each quotient by a subhypergroup carries a hypergroup structure
of the
same type.
The algebraic properties are partially related to properties of
random walks
on matrix Bessel hypergroups. In particular, known properties of Wishart
distributions, which form Gaussian convolution semigroups on these
hypergroups,
are put into a new light. Moreover, limit theorems for random walks
on these
hypergroups are presented. In particular, we obtain strong laws of large
numbers and a central limit theorem with Wishart distributions as
limits.
http://front.math.ucdavis.edu/math.CA/0603017
---------------------------------------------------------------
4170. LINEAR STOCHASTIC DIFFERENTIAL EQUATIONS DRIVEN BY A FRACTIONAL
BROWNIAN MOTION WITH HURST PARAMETER LESS THAN 1/2
Jorge A. Leon and Jaime San Martin
In this paper we use the chaos decomposition approach to establish the
existence of a unique continuous solution to linear fractional
differential
equations of the Skorohod type. Here the coefficients are
deterministic, the
inital condition is anticipating and the underlying fractional
Brownian motion
has Hurst parameter less than 1/2. We provide an explicit expression
for the
chaos decomposition of the solution in order to show our results.
http://front.math.ucdavis.edu/math.PR/0603636
---------------------------------------------------------------
4171. LIFETIME ASYMPTOTICS OF ITERATED BROWNIAN MOTION IN R^{N}
Erkan Nane
Let $\tau_{D}(Z) $ be the first exit time of iterated Brownian motion
from a
domain $D \subset \RR{R}^{n}$ started at $z\in D$ and let $P_{z}[\tau_
{D}(Z)
>t]$ be its distribution. In this paper we establish the exact
asymptotics of
$P_{z}[\tau_{D}(Z) >t]$ over bounded domains as an improvement of the
results
in \cite{deblassie, nane2}, for $z\in D$ \begin{eqnarray}
\lim_{t\to\infty} t^{-1/2}\exp({3/2}\pi^{2/3}\lambda_{D}^{2/3}t^
{1/3})
P_{z}[\tau_{D}(Z)>t]= C(z),\nonumber \end{eqnarray} where
$C(z)=(\lambda_{D}2^{7/2})/\sqrt{3 \pi}(\psi(z)\int_{D}\psi(y)dy) ^{2}
$. Here
$\lambda_{D}$ is the first eigenvalue of the Dirichlet Laplacian
${1/2}\Delta$
in $D$, and $\psi $ is the eigenfunction corresponding to $\lambda_{D}
$ .
We also study lifetime asymptotics of Brownian-time Brownian
motion (BTBM),
$Z^{1}_{t}=z+X(|Y(t)|)$, where $X_{t}$ and $Y_{t}$ are independent
one-dimensional Brownian motions.
http://front.math.ucdavis.edu/math.PR/0603637
---------------------------------------------------------------
4172. EDGEWORTH EXPANSION OF THE LARGEST EIGENVALUE DISTRIBUTION
FUNCTION OF GUE AND LUE
Leonard N. Choup
We derive expansions of the Hermite and Laguerre kernels at the edge
of the
spectrum of the finite n Gaussian Unitary Ensemble (GUEn) and the
finite n
Laguerre Unitary Ensem- ble (LUEn), respectively. Using these large n
kernel
expansions, we prove an Edgeworth type theorem for the largest
eigenvalue
distribution function of GUEn and LUEn. In our Edgeworth expansion, the
correction terms are expressed in terms of the same Painleve II function
appearing in the leading term, i.e. in the Tracy-Widom distribution. We
conclude with a brief discussion of the universality of these results.
http://front.math.ucdavis.edu/math.PR/0603639
---------------------------------------------------------------
4173. THE METASTABILITY THRESHOLD FOR MODIFIED BOOTSTRAP PERCOLATION
IN D DIMENSIONS
Alexander E. Holroyd
In the modified bootstrap percolation model, sites in the cube
{1,...,L}^d
are initially declared active independently with probability p. At
subsequent
steps, an inactive site becomes active if it has at least one active
nearest
neighbour in each of the d dimensions, while an active site remains
active
forever. We study the probability that the entire cube is eventually
active.
For all d>=2 we prove that as L\to\infty and p\to 0 simultaneously, this
probability converges to 1 if L=exp^{d-1} (lambda+epsilon)/p, and
converges to
0 if L=exp^{d-1} (lambda-epsilon)/p, for any epsilon>0. Here exp^n
denotes the
n-th iterate of the exponential function, and the threshold lambda
equals
pi^2/6 for all d.
http://front.math.ucdavis.edu/math.PR/0603645
---------------------------------------------------------------
4174. LOG-CONCAVITY AND THE MAXIMUM ENTROPY PROPERTY OF THE POISSON
DISTRIBUTION
Oliver Johnson
We prove that the Poisson distribution maximises entropy in the class of
ultra-log-concave distributions, extending a result of Harremo\"{e}s.
The proof
uses ideas concerning log-concavity, and a semigroup action involving
adding
Poisson variables and thinning. We go on to show that the entropy is
a concave
function along this semigroup.
http://front.math.ucdavis.edu/math.PR/0603647
---------------------------------------------------------------
4175. QUENCHED NONEQUILIBRIUM CENTRAL LIMIT THEOREM FOR A TAGGED
PARTICLE IN THE EXCLUSION PROCESS WITH BOND DISORDER
M. D. Jara and C. Landim
For a sequence of i.i.d. random variables $\{\xi_x : x\in \bb Z\}$
bounded
above and below by strictly positive finite constants, consider the
nearest-neighbor one-dimensional simple exclusion process in which a
particle
at $x$ (resp. $x+1$) jumps to $x+1$ (resp. $x$) at rate $\xi_x$. We
examine a
quenched nonequilibrium central limit theorem for the position of a
tagged
particle in the exclusion process with bond disorder $\{\xi_x : x\in
\bb Z\}$.
We prove that the position of the tagged particle converges under
diffusive
scaling to a Gaussian process if the other particles are initially
distributed
according to a Bernoulli product measure associated to a smooth profile
$\rho_0:\bb R\to [0,1]$.
http://front.math.ucdavis.edu/math.PR/0603653
---------------------------------------------------------------
4176. ON DECOMPOSING RISK IN A FINANCIAL-INTERMEDIATE MARKET AND
RESERVING
Saul Jacka and Abdel Berkaoui
We consider the problem of decomposing monetary risk in the presence
of a
fully traded market in {\it some} risks. We show that a mark-to-
market approach
to pricing leads to such a decomposition if the risk measure is time-
consistent
in the sense of Delbaen.
http://front.math.ucdavis.edu/math.PR/0603041
---------------------------------------------------------------
4177. ERGODIC THEORY FOR SDES WITH EXTRINSIC MEMORY
M. Hairer and A. Ohashi
We develop a theory of ergodicity for a class of random dynamical
systems
where the driving noise is not white. The two main tools of our
analysis are
the strong Feller property and topological irreducibility, introduced
in this
work for a class of non-Markovian systems. They allow us to obtain a
criteria
for ergodicity which is similar in nature to the Doob-Khas'minskii
theorem.
The second part of this article shows how it is possible to apply
these
results to the case of stochastic differential equations driven by
fractional
Brownian motion. It follows that under a non-degeneracy condition on
the noise,
such equations admit a unique adapted stationary solution.
http://front.math.ucdavis.edu/math.PR/0603658
---------------------------------------------------------------
4178. CORRECTION. CONNECT THE DOTS: HOW MANY RANDOM POINTS CAN A
REGULAR CURVE PASS THROUGH?
E. Arias-Castro and D. L. Donoho and X. Huo and C. A. Tovey
Correction for Adv. in Appl. Probab. 37, no. 3 (2005), 571-603
http://front.math.ucdavis.edu/math.PR/0603673
---------------------------------------------------------------
4179. LARGE DEVIATIONS FOR MANY BROWNIAN BRIDGES WITH SYMMETRISED
INITIAL-TERMINAL CONDITION
Stefan Adams and Wolfgang K\"onig
Consider a large system of $N$ Brownian motions in $\mathbb{R}^d$
with some
non-degenerate initial measure on some fixed time interval $[0,\beta]
$ with
symmetrised initial-terminal condition. That is, for any $i$, the
terminal
location of the $i$-th motion is affixed to the initial point of the
$\sigma(i)$-th motion, where $\sigma$ is a uniformly distributed random
permutation of $1,...,N$. Such systems play an important role in quantum
physics in the description of Boson systems at positive temperature
$1/\beta$.
In this paper, we describe the large-N behaviour of the empirical
path
measure (the mean of the Dirac measures in the $N$ paths) and of the
mean of
the normalised occupation measures of the $N$ motions in terms of large
deviations principles. The rate functions are given as variational
formulas
involving certain entropies and Fenchel-Legendre transforms.
Consequences are
drawn for asymptotic independence statements and laws of large numbers.
In the special case related to quantum physics, our rate function
for the
occupation measures turns out to be equal to the well-known Donsker-
Varadhan
rate function for the occupation measures of one motion in the limit of
diverging time. This enables us to prove a simple formula for the
large-N
asymptotic of the symmetrised trace of ${\rm e}^{-\beta \mathcal{H}_N}
$, where
$\mathcal{H}_N$ is an $N$-particle Hamilton operator in a trap.
http://front.math.ucdavis.edu/math.PR/0603702
---------------------------------------------------------------
4180. FINITELY ADDITIVE BELIEFS AND UNIVERSAL TYPE SPACES
Martin Meier
The probabilistic type spaces in the sense of Harsanyi [Management
Sci. 14
(1967/68) 159--182, 320--334, 486--502] are the prevalent models used to
describe interactive uncertainty. In this paper we examine the
existence of a
universal type space when beliefs are described by finitely additive
probability measures. We find that in the category of all type spaces
that
satisfy certain measurability conditions ($\kappa$-measurability, for
some
fixed regular cardinal $\kappa$), there is a universal type space
(i.e., a
terminal object) to which every type space can be mapped in a unique
beliefs-preserving way. However, by a probabilistic adaption of the
elegant
sober-drunk example of Heifetz and Samet [Games Econom. Behav. 22 (1998)
260--273] we show that if all subsets of the spaces are required to be
measurable, then there is no universal type space.
http://front.math.ucdavis.edu/math.PR/0602656
---------------------------------------------------------------
4181. THE TIME EVOLUTION OF PERMUTATIONS UNDER RANDOM STIRRING
B\'alint Vet\H{o}
We consider permutations of $\{1,...,n\}$ obtained by $\sqrt{nt}$
independent
applications of random stirring. In each step the same marked
stirring element
is transposed with probability $1/n$ with any one of the $n$ elements.
Normalizing by $\sqrt{n}$ we describe the asymptotic distribution of
the cycle
structure of these permutations, for all $t\ge0$, as $n\to\infty$.
http://front.math.ucdavis.edu/math.PR/0603044
---------------------------------------------------------------
4182. STATIONARITY OF PURE DELAY SYSTEMS AND QUEUES WITH IMPATIENT
CUSTOMERS VIA STOCHASTIC RECURSIONS
Pascal Moyal
In this paper we solve a particular stochastic recursion in the
stationary
ergodic framework, and propose some applications of this result to
the study of
regenerativity (that is, finiteness of busy cycles) and stationarity
of some
queueing systems: pure delay systems, in which all customers are
immediately
served, and queues with impatient customers. In this latter case
under the FIFO
discipline, we prove as well the existence of a stationary workload
on an
enlarged probability space.
http://front.math.ucdavis.edu/math.PR/0603709
---------------------------------------------------------------
4183. ON THE ASYMPTOTIC DISTRIBUTION OF CERTAIN BIVARIATE REINSURANCE
TREATIES
Enkelejd Hashorva
Let (X_n,Y_n), n\ge 1 be bivariate random claim sizes with common
distribution function F and let N(t), t \ge 0 be a stochastic process
which
counts the number of claims that occur in the time interval [0,t], t
\ge 0. In
this paper we derive the joint asymptotic distribution of randomly
indexed
order statistics of the random sample
(X_1,Y_1),(X_2,Y_2),...,(X_{N(t)},Y_{N(t)}) which is then used to obtain
asymptotic representations for the joint distribution of two generalised
largest claims reinsurance treaties available under specific insurance
settings. As a by-product we obtain a stochastic representation of a
m-dimensional Lambda-extremal variate in terms of iid unit
exponential random
variables.
http://front.math.ucdavis.edu/math.PR/0603719
---------------------------------------------------------------
4184. THE ZEROS OF GAUSSIAN RANDOM HOLOMORPHIC FUNCTIONS ON $\C^N$,
AND HOLE PROBABILITY
Scott Zrebiec
We consider a class of Gaussian random holomorphic functions, whose
expected
zero set is uniformly distributed over $\C^n $. This class is unique
(up to
multiplication by a non zero holomorphic function), and is closely
related to a
Gaussian field over a Hilbert space of holomorphic functions on the
reduced
Heisenberg group. For a fixed random function of this class, we show
that the
probability that there are no zeros in a ball of large radius, is
less than
$e^{-c_1 r^{2n+2}}$, and is also greater than $e^{-c_2 r^{2n+2}}$.
Enroute to
this result we also compute probability estimates for the event that
a random
function's unintegrated counting function deviates significantly from
its mean.
http://front.math.ucdavis.edu/math.CV/0603696
---------------------------------------------------------------
4185. EXCHANGEABLE PARTITIONS DERIVED FROM MARKOVIAN COALESCENTS
Rui Dong and Alexander Gnedin and Jim Pitman
Kingman derived the Ewens sampling formula for random partitions
describing
the genetic variation in a neutral mutation model defined by a
Poisson process
of mutations along lines of descent governed by a simple coalescent
process,
and observed that similar methods could be applied to more complex
models.
M{\"o}hle described the recursion which determines the generalization
of the
Ewens sampling formula in the situation when the lines of descent are
governed
by a $\Lambda$-coalescent, which allows multiple mergers. Here we
show that the
basic integral representation of transition rates for the $\Lambda$-
coalescent
is forced by sampling consistency under more general assumptions on the
coalescent process. Exploiting an analogy with the theory of
regenerative
partition structures, we provide various characterizations of the
associated
partition structures in terms of discrete-time Markov chains.
http://front.math.ucdavis.edu/math.PR/0603745
---------------------------------------------------------------
4186. BEHAVIOR OF THE EULER SCHEME WITH DECREASING STEP IN A
DEGENERATE SITUATION
Vincent Lemaire (LAMA)
The aim of this paper is to study the behavior of the weighted empirical
measures of the decreasing step Euler scheme of a one-dimensional
diffusion
process having multiple invariant measures. This situation can occur
when the
drift and the diffusion coefficient are vanish simultaneously. As a
first step,
we give a brief description of the Feller's classification of the
one-dimensional process. We recall the concept of attractive and
repulsive
boundary point and introduce the concept of strongly repulsive point.
That
allows us to establish a classification of the ergodic behavior of the
diffusion. We conclude this section by giving necessary and sufficient
conditions on the nature of boundary points in terms of Lyapunov
functions. In
the second section we use this characterization to study the
decreasing step
Euler scheme. We give also an numerical example in higher dimension.
http://front.math.ucdavis.edu/math.PR/0604021
---------------------------------------------------------------
4187. INVASION AND ADAPTIVE EVOLUTION FOR INDIVIDUAL-BASED SPATIALLY
STRUCTURED POPULATIONS
Nicolas Champagnat (WIAS) and Sylvie M\'{e}l\'{e}ard (MODAL'X and
FESE)
The interplay between space and evolution is an important issue in
population
dynamics, that is in particular crucial in the emergence of
polymorphism and
spatial patterns. Recently, biological studies suggest that invasion and
evolution are closely related. Here we model the interplay between
space and
evolution starting with an individual-based approach and show the
important
role of parameter scalings on clustering and invasion. We consider a
stochastic
discrete model with birth, death, competition, mutation and spatial
diffusion,
where all the parameters may depend both on the position and on the
trait of
individuals. The spatial motion is driven by a reflected diffusion in
a bounded
domain. The interaction is modelled as a trait competition between
individuals
within a given spatial interaction range. First, we give an algorithmic
construction of the process. Next, we obtain large population
approximations,
as weak solutions of nonlinear reaction-diffusion equations with
Neumann's
boundary conditions. As the spatial interaction range is fixed, the
nonlinearity is nonlocal. Then, we make the interaction range
decrease to zero
and prove the convergence to spatially localized nonlinear reaction-
diffusion
equations, with Neumann's boundary conditions. Finally, simulations
based on
the microscopic individual-based model are given, illustrating the
strong
effects of the spatial interaction range on the emergence of spatial and
phenotypic diversity (clustering and polymorphism) and on the
interplay between
invasion and evolution. The simulations focus on the qualitative
differences
between local and nonlocal interactions.
http://front.math.ucdavis.edu/math.PR/0604041
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4188. PROCESSES WITH INERT DRIFT
David White
We construct a stochastic process whose drift is a function of the
process's
local time at a reflecting barrier. The process arose as a model of the
interactions of a Brownian particle and an inert particle in \citep
{knight:01}.
Interesting asymptotic results are obtained for two different
arrangements of
inert particles and Brownian particles. A version of the process in $
\Re^d$ is
also constructed.
http://front.math.ucdavis.edu/math.PR/0604052
---------------------------------------------------------------
4189. WHEN THE LAW OF LARGE NUMBERS FAILS FOR INCREASING SUBSEQUENCES
OF RANDOM PERMUTATIONS
Ross G. Pinsky
Let the random variable $Z_{n,k}$ denote the number of increasing
subsequences of length $k$ in a random permutation from $S_n$, the
symmetric
group of permutations of $\{1,...,n\}$. In a recent paper
(http://front.math.ucdavis.edu/math.PR/0407353) we showed that the
weak law of
large numbers holds for $Z_{n,k_n}$ if $k_n=o(n^\frac25)$; that is, $$
\lim_{n\to\infty}\frac{Z_{n,k_n}} {EZ_{n,k_n}}=1, \text{in
probability}. $$ The
method of proof employed there used the second moment method and
demonstrated
that this method cannot work if the condition $k_n=o(n^\frac25)$ does
not hold.
It follows from results concerning the longest increasing subsequence
of a
random permutation that the law of large numbers cannot hold for $Z_
{n,k_n}$ if
$k_n\ge cn^\frac12$, with $c>2$. Presumably there is a critical
exponent $l_0$
such that the law of large numbers holds if $k_n=O(n^l)$, with $l<l_0
$, and
does not hold if $\limsup_{n\to\infty}\frac{k_n}{n^l}>0$, for some
$l>l_0$.
Several phase transitions concerning increasing subsequences occur at
$l=\frac12$, and these would suggest that $l_0=\frac12$. However, in
this
paper, we show that the law of large numbers fails for $Z_{n,k_n}$ if
$\limsup_{n\to\infty}\frac{k_n}{n^\frac49}=\infty$. Thus the critical
exponent,
if it exists, must satisfy $l_0\in[\frac25,\frac49]$.
http://front.math.ucdavis.edu/math.PR/0604067
---------------------------------------------------------------
4190. A SIMPLE FLUCTUATION LOWER BOUND FOR A DISORDERED MASSLESS
RANDOM CONTINUOUS SPIN MODEL IN D=2
C. Kuelske and E. Orlandi
We prove a finite volume lower bound of the order of the squareroot
of log N
on the delocalization of a disordered continuous spin model (resp.
effective
interface model) in d = 2 in a box of size N . The interaction is
assumed to be
massless, possibly anharmonic and dominated from above by a Gaussian.
Disorder
is entering via a linear source term. For this model delocalization
with the
same rate is proved to take place already without disorder, so our
proof shows
that randomness will only enhance fluctuations.
http://front.math.ucdavis.edu/math.PR/0604068
---------------------------------------------------------------
4191. THE MAXIMUM OF THE LOCAL TIME OF A DIFFUSION PROCESS IN A
DRIFTED BROWNIAN POTENTIAL
Alexis Devulder (PMA)
We consider a one-dimensional diffusion process in a drifted Brownian
potential. We are interested in the maximum of its local time, and
study its
almost sure asymptotic behaviour, which is proved to be different
from the
behaviour of the maximum local time of the transient random walk in
random
environment.
http://front.math.ucdavis.edu/math.PR/0604078
---------------------------------------------------------------
4192. CAVITY METHOD IN THE SPHERICAL SK MODEL
Dmitry Panchenko
We develop the cavity method for the spherical Sherrington-
Kirkpatrick model
at high temperature and small external field. As one application, we
carry out
the second moment computations for the overlap and the magnetization.
http://front.math.ucdavis.edu/math.PR/0604081
---------------------------------------------------------------
4193. ON THE OVERLAP IN THE MULTIPLE SPHERICAL SK MODELS
Dmitry Panchenko and Michel Talagrand
In order to study certain questions concerning the distribution of the
overlap in Sherrington-Kirkpatrick type models, such as the chaos and
ultrametricity problems, it seems natural to study the free energy of
multiple
systems with constrained overlaps. One can write analogues of
Guerra's replica
symmetry breaking bound for such systems but it is not at all obvious
how to
choose informative functional order parameters in these bounds. We
were able to
make some progress for spherical pure $p$-spin SK models where many
computations can be made explicitly. For pure 2-spin model we prove
ultrametricity and chaos in an external field. For the pure $p$-spin
model for
even $p>4$ without an external field we describe two possible values
of the
overlap of two systems at different temperatures. We also prove a
somewhat
unexpected result which shows that in the 2-spin model the support of
the joint
overlap distribution is not always witnessed at the level of the free
energy
and,for example, ultrametricity holds only in a weak sense.
http://front.math.ucdavis.edu/math.PR/0604082
---------------------------------------------------------------
4194. DERIVATIVES OF ENTROPY RATE IN SPECIAL FAMILIES OF HIDDEN
MARKOV CHAINS
Guangyue Han and Brian Marcus
Consider a hidden Markov chain obtained as the observation process of an
ordinary Markov chain corrupted by noise. Zuk, et. al. [13], [14]
showed how,
in principle, one can explicitly compute the derivatives of the
entropy rate of
at extreme values of the noise. Namely, they showed that the
derivatives of
standard upper approximations to the entropy rate actually stabilize
at an
explicit finite time. We generalize this result to a natural class of
hidden
Markov chains called ``Black Holes.'' We also discuss in depth
special cases of
binary Markov chains observed in binary symmetric noise, and give an
abstract
formula for the first derivative in terms of a measure on the simplex
due to
Blackwell.
http://front.math.ucdavis.edu/cs.IT/0603059
---------------------------------------------------------------
4195. THE CODING OF COMPACT REAL TREES BY REAL VALUED FUNCTIONS
Thomas Duquesne
This paper is a detailled study of the coding of real trees by real
valued
functions that is motivated by probabilistic problems related to
continuum
random trees. Indeed it is known since the works of Aldous (1993) and
Le Gall
(1991) that a continuous non-negative function $h$ on $[0,1]$ such that
$h(0)=0$ can be seen as the contour process of a compact real tree. This
particular coding of a compact real tree provides additional
structures, namely
a root that is the vertex corresponding to $0\in [0,1]$, a linear order
inherited from the usual order on $[0,1]$ and a measure induced by
the Lebesgue
measure on $[0,1]$; of course, the root, the linear order and the
measure
obtained by such a coding have to satisfy some compatibility
conditions. In
this paper, we prove that any compact real tree equipped with a root,
a linear
order and a measure that are compatible can be encoded by a non-negative
function $h$ defined on a finite interval $[0, M]$, that is assumed
to be
left-continuous with right-limit, without positive jump and such that
$h(0+)=h(0)=0$. Moreover, this function is unique if we assume that the
exploration of the tree induced by such a coding backtracks as less as
possible. We also prove that a measure-change on the tree corresponds
to a
re-parametrization of the coding function. In addition, we describe
several
path-properties of the coding function in terms of the metric
properties of the
real tree.
http://front.math.ucdavis.edu/math.PR/0604106
---------------------------------------------------------------
4196. ON THE FUTURE INFIMUM OF POSITIVE SELF-SIMILAR MARKOV PROCESSES
J.C. Pardo
We establish integral tests and laws of the iterated logarithm for
the upper
envelope of the future infimum of positive self-similar Markov
processes and
for increasing self-similar Markov processes at 0 and infinity. Our
proofs are
based on the Lamperti representation and time reversal arguments due to
Chaumont and Pardo [9]. These results extend laws of the iterated
logarithm for
the future infimum of Bessel processes due to Khoshnevisan et al. [11].
http://front.math.ucdavis.edu/math.PR/0604110
---------------------------------------------------------------
4197. LAWS AND LIKELIHOODS FOR ORNSTEIN UHLENBECK-GAMMA AND OTHER BNS
OU STOCHASTIC VOLATILTY MODELS WITH EXTENSIONS
Lancelot F. James
In recent years there have been many proposals as flexible
alternatives to
Gaussian based continuous time stochastic volatility models. A great
deal of
these models employ positive L\'evy processes. Among these are the
attractive
non-Gaussian positive Ornstein-Uhlenbeck (OU) processes proposed by
Barndorff-Nielsen and Shephard (BNS) in a series of papers. One
current problem
of these approaches is the unavailability of a tractable likelihood
based
statistical analysis for the returns of financial assets. This paper,
while
focusing on the BNS models, develops general theory for the
implementation of
statistical inference for a host of models. Specifically we show how
to reduce
the infinite-dimensional process based models to finite, albeit high,
dimensional ones. Inference can then be based on Monte Carlo methods. As
highlights, specific to BNS we show that an OU process driven by an
infinite
activity Gamma process, that is an OU-$\Gamma$, exhibits unique
features which
allows one to exactly sample from relevant joint distributions. We
show that
this is a consequence of the OU structure and the unique calculus of
Gamma and
Dirichlet processes. Owing to another connection between Gamma/Dirichlet
processes and the theory of Generalized Gamma Convolutions (GGC) we
identify a
large class of models, we call (FGGC), where one can perfectly sample
marginal
distributions relevant to option pricing and Monte Carlo likelihood
analysis.
This involves a curious result, we establish as Theorem 6.1. We also
discuss
analytic techniques and candidate densities for Monte-Carlo
procedures which
can be applied to more general
http://front.math.ucdavis.edu/math.ST/0604086
---------------------------------------------------------------
4198. MAXIMUM PRINCIPLE FOR SPDES AND ITS APPLICATIONS
N.V. Krylov
The maximum principle for SPDEs is established in multidimensional $C^
{1}$
domains. An application is given to proving the H\"older continuity
up to the
boundary of solutions of one-dimensional SPDEs.
http://front.math.ucdavis.edu/math.PR/0604125
---------------------------------------------------------------
4199. A FAMILY OF NON-GAUSSIAN MARTINGALES WITH GAUSSIAN MARGINALS
kais Hamza and Fima C. Klebaner
We construct a family of non-Gaussian martingales the marginals of
which are
all Gaussian. We give the predictable quadratic variation of these
processes
and show they do not have continuous paths. These processes are
Markovian and
inhomogeneous in time, and we give their infinitesimal generators.
Within this
family we find a class of piecewise deterministic pure jump processes
and
describe the laws of jumps and times between the jumps.
http://front.math.ucdavis.edu/math.PR/0604127
---------------------------------------------------------------
4200. STOCHASTIC EQUATIONS WITH TIME-DEPENDENT DRIFT DRIVEN BY LEVY
PROCESSES
V.P.Kurenok
Using the method of Krylov's estimates, we prove the existence of weak
solutions of stochastic differential equations driven by purely
discontinuous
Levy processes satisfying an additional assumption. The diffusion
coefficient
is assumed to be one and the time-dependent drift is measurable and
bounded.
http://front.math.ucdavis.edu/math.PR/0604136
---------------------------------------------------------------
4201. CONDITIONED GALTON-WATSON TREES DO NOT GROW
Svante Janson
An example is given which shows that, in general, conditioned Galton-
Watson
trees cannot be obtained by adding vertices one by one, as has been
shown in a
special case by Luczak and Winkler.
http://front.math.ucdavis.edu/math.PR/0604141
---------------------------------------------------------------
4202. SEMI-SELFDECOMPOSABLE LAWS IN THE MINIMUM SCHEME
S Satheesh and E Sandhya
We discuss semi-selfdecomposable laws in the minimum scheme and
characterize
them using an autoregressive model. Semi-Pareto and semi-Weibull laws
of Pillai
(1991) are shown to be semi-selfdecomposable in this scheme. Methods for
deriving this class of laws are then attempted from the angle of
randomization.
Finally, discrete analogues of these results are also considered.
http://front.math.ucdavis.edu/math.PR/0604146
---------------------------------------------------------------
4203. UNIFORM FORMULAE FOR COEFFICIENTS OF MEROMORPHIC FUNCTIONS IN
TWO VARIABLES. PART I
Manuel Lladser
Uniform asymptotic formulae for arrays of complex numbers of the form
$(f_{r,s})$, with $r$ and $s$ nonnegative integers, are provided as $r
$ and $s$
converge to infinity at a comparable rate. Our analysis is restricted
to the
case in which the generating function $F(z,w):=\sum f_{r,s} z^r w^s$ is
meromorphic in a neighborhood of the origin. We provide uniform
asymptotic
formulae for the coefficients $f_{r,s}$ along directions in the $(r,s)
$-lattice
determined by regular points of the singular variety of $F$. Our main
result
derives from the analysis of a one dimensional parameter-varying
integral
describing the asymptotic behavior of $f_{r,s}$. We specifically
consider the
case in which the phase term of this integral has a unique stationary
point,
however, allowing the possibility that one or more stationary points
of the
amplitude term coalesce with this. Our results find direct
application in
certain problems associated to the Lagrange inversion formula as well as
bivariate generating functions of the form $v(z)/(1-w\cdot u(z))$.
http://front.math.ucdavis.edu/math.CO/0604152
---------------------------------------------------------------
4204. DESIGN FLAWS IN THE IMPLEMENTATION OF THE ZIGGURAT AND MONTY
PYTHON METHODS (AND SOME REMARKS ON MATLAB RANDN)
Boaz Nadler
{\em Ziggurat} and {\em Monty Python} are two fast and elegant methods
proposed by Marsaglia and Tsang to transform uniform random variables
to random
variables with normal, exponential and other common probability
distributions.
While the proposed methods are theoretically correct, we show that
there are
various design flaws in the uniform pseudo random number generators
(PRNG's) of
their published implementations for both the normal and Gamma
distributions
\cite{Ziggurat,{Gamma},Monty}. These flaws lead to non-uniformity of the
resulting pseudo-random numbers and consequently to noticeable
deviations of
their outputs from the required distributions. In addition, we show
that the
underlying uniform PRNG of the published implementation of Matlab's
\texttt{randn}, which is also based on the Ziggurat method, is not
uniformly
distributed with correlations between consecutive pairs. Also, we
show that the
simple linear initialization of the registers in matlab's \texttt
{randn} may
lead to non-trivial correlations between output sequences initialized
with
different (related or even random unrelated) seeds. These, in turn,
may lead to
erroneous results for stochastic simulations.
http://front.math.ucdavis.edu/math.ST/0603058
---------------------------------------------------------------
4205. EFFECTIVE BANDWIDTH PROBLEM REVISITED
Vyacheslav M. Abramov
The paper studies a single-server queueing system with autonomous
service and
$\ell$ priority classes. Arrival and departure processes are defined
by marked
point processes. There are $\ell$ buffers corresponding to priority
classes,
and upon arrival a unit of the $k$th priority class occupies the
place in the
$k$th buffer. Let $N^{(k)}$, $k=1,2,...,\ell$ denote the quota for
the total
$k$th buffer content. The values $N^{(k)}$ are assumed to be large, and
queueing systems both with finite and infinite buffers are studied.
In the case
of system with finite buffers, the values $N^{(k)}$ characterize buffer
capacities.
The paper discusses a circle of problems related to optimization of
performance measures associated with overflowing the quota of buffer
contents.
Our approach to this problem is new, and presentation of our results
is simple
and clear for real applications.
http://front.math.ucdavis.edu/math.PR/0604182
---------------------------------------------------------------
4206. LIMITING BEHAVIOR OF THE DISTANCE OF A RANDOM WALK
Nathanael Berestycki and Rick Durrett
This investigation is motivated by a result we proved recently for
the random
transposition random walk: the distance from the starting point of
the walk has
a phase transition from a linear regime to a sublinear regime at time
$n/2$.
Here, we study three new examples. It is trivial that the distance
for random
walk on the hypercube is smooth and is given by one simple formula.
In the case
of random adjacent transpositions, we find that there is no phase
transition
even though the distance has different scalings in three different
regimes. In
the case of a random 3-regular graph, there is a phase transition
from linear
growth to a constant equal to the diameter of the graph, at time $3
\log_2 n$.
http://front.math.ucdavis.edu/math.PR/0604188
---------------------------------------------------------------
4207. HEAVY TAILS IN LAST-PASSAGE PERCOLATION
Ben Hambly and James B. Martin
We consider last-passage percolation models in two dimensions, in
which the
underlying weight distribution has a heavy tail of index alpha<2. We
prove
scaling laws and asymptotic distributions, both for the passage times
and for
the shape of optimal paths; these are expressed in terms of a family
(indexed
by alpha) of "continuous last-passage percolation" models in the unit
square.
In the extreme case alpha=0 (corresponding to a distribution with slowly
varying tail) the asymptotic distribution of the optimal path can be
represented by a random self-similar measure on [0,1], whose
multifractal
spectrum we compute. By extending the continuous last-passage
percolation model
to R^2 we obtain a heavy-tailed analogue of the Airy process,
representing the
limit of appropriately scaled vectors of passage times to different
points in
the plane. We give corresponding results for a directed percolation
problem
based on alpha-stable Levy processes, and indicate extensions of the
results to
higher dimensions.
http://front.math.ucdavis.edu/math.PR/0604189
---------------------------------------------------------------
4208. TWO NON-REGULAR EXTENSIONS OF LARGE DEVIATION BOUND
Masahito Hayashi
We formulate two types of extensions of the large deviation theory
initiated
by Bahadur in a non-regular setting. One can be regarded as a bound
of the
point estimation, the other can be regarded as the limit of a bound
of the
interval estimation. Both coincide in the regular case, but do not
necessarily
coincide in a non-regular case. Using the limits of relative R\'{e}nyi
entropies, we derive their upper bounds and give a necessary and
sufficient
condition for the coincidence of the two upper bounds. We also
discuss the
attainability of these two bounds in several non-regular location shift
families.
http://front.math.ucdavis.edu/math.PR/0604197
---------------------------------------------------------------
4209. ATTRACTING EDGE AND STRONGLY EDGE REINFORCED WALKS
V. Limic and P. Tarres
The goal is to show that an edge reinforced random walk on a graph of
bounded
degree, with reinforcement {\em weight function} $W$ taken from a
general class
of reciprocally summable reinforcement weight functions, traverses a
random
{\em attracting} edge at all large times.
The statement of the main theorem is very close to settling the
original
conjecture of Sellke (1994). An important corollary of this main
result says
that if $W$ is reciprocally summable and nondecreasing, the
attracting edge
exists on any graph of bounded degree, with probability 1. Another
corollary is
the main theorem of Limic (2003) where the class of weights was
restricted to
reciprocally summable powers.
The proof uses martingale and other techniques developed by the
authors in
separate studies of edge and vertex reinforced walks (Limic (2003),
Tarr\`es
(2004)), and of nonconvergence properties of stochastic algorithms
towards
unstable equilibrium points of the associated deterministic dynamics,
Tarr\`es
(2000).
http://front.math.ucdavis.edu/math.PR/0604200
---------------------------------------------------------------
4210. THE MOMENT PROBLEM AND THE WIENER SPACE
Frederik S Herzberg
Consider an $L^1$-continuous functional $\ell$ on the vector space of
polynomials of Brownian motion at given times, suppose $\ell $
commutes with
the quadratic variation in a natural sense, and consider a finite set of
polynomials of Brownian motion at rational times, $p_1(\vec
b),...,p_m,(\vec
b)$, mapping the Wiener space to $\mathbb{R}$.
Similarly to the moment problem for a finite-dimensional space of
polynomials, we give sufficient conditions under which $\ell$ can be
written in
the form $\int \cdot d\mu$ for some finite measure $\mu$ on the
Wiener space
such that $\mu$-almost surely, all the random variables $p_1(\vec
b),...,p_m,(\vec b)$ are nonnegative.
http://front.math.ucdavis.edu/math.PR/0604211
---------------------------------------------------------------
4211. PROCESSOR SHARING QUEUES WITH IMPATIENCE
Christian H. Gromoll (STANFORD-MATHS) and Philippe Robert (INRIA
Rocquencourt), Bert Zwart (TUE)
We investigate a processor sharing queue with renewal arrivals and
generally
distributed service times. Impatient jobs may abandon the queue, or
renege,
before completing service. The corresponding stochastic processes are
represented by measure valued Markov processes on R^2\_+. A scaling
procedure
that gives rise to a fluid model with a nontrivial, yet tractable
steady state
behavior, is presented. This fluid model model captures many
essential features
of the underlying stochastic model, and it is used to analyze the
impact of
impatience in processor sharing queues.
http://front.math.ucdavis.edu/math.PR/0604215
---------------------------------------------------------------
4212. HEURISTICS FOR THE WHITEHEAD MINIMIZATION PROBLEM
R.M. Haralick and A.D. Miasnikov and A.G. Myasnikov
In this paper we discuss several heuristic strategies which allow one to
solve the Whitehead's minimization problem much faster (on most
inputs) than
the classical Whitehead algorithm. The mere fact that these
strategies work in
practice leads to several interesting mathematical conjectures. In
particular,
we conjecture that the length of most non-minimal elements in a free
group can
be reduced by a Nielsen automorphism which can be identified by
inspecting the
structure of the corresponding Whitehead Graph.
http://front.math.ucdavis.edu/math.GR/0604204
---------------------------------------------------------------
4213. RIGOROUS INEQUALITIES BETWEEN LENGTH AND TIME SCALES IN GLASSY
SYSTEMS
Andrea Montanari and Guilhem Semerjian
Glassy systems are characterized by an extremely sluggish dynamics
without
any simple sign of long range order. It is a debated question whether
a correct
description of such phenomenon requires the emergence of a large
correlation
length. We prove rigorous bounds between length and time scales
implying the
growth of a properly defined length when the relaxation time
increases. Our
results are valid in a rather general setting, which covers finite-
dimensional
and mean field systems.
As an illustration, we discuss the Glauber (heat bath) dynamics of
p-spin
glass models on random regular graphs. We present the first proof
that a model
of this type undergoes a purely dynamical phase transition not
accompanied by
any thermodynamic singularity.
http://front.math.ucdavis.edu/cond-mat/0603018
---------------------------------------------------------------
4214. A HYBRID SEARCH ALGORITHM FOR THE WHITEHEAD MINIMIZATION PROBLEM
A.D. Myasnikov and R.M Haralick
The Whitehead Minimization problem is a problem of finding elements
of the
minimal length in the automorphic orbit of a given element of a free
group. The
classical algorithm of Whitehead that solves the problem depends
exponentially
on the group rank. Moreover, it can be easily shown that exponential
blowout
occurs when a word of minimal length has been reached and, therefore, is
inevitable except for some trivial cases.
In this paper we introduce a deterministic Hybrid search algorithm
and its
stochastic variation for solving the Whitehead minimization problem.
Both
algorithms use search heuristics that allow one to find a length-
reducing
automorphism in polynomial time on most inputs and significantly
improve the
reduction procedure. The stochastic version of the algorithm employs a
probabilistic system that decides in polynomial time whether or not a
word is
minimal. The stochastic algorithm is very robust. It has never
happened that a
non-minimal element has been claimed to be minimal.
http://front.math.ucdavis.edu/math.GR/0604206
---------------------------------------------------------------
4215. ANALYSE NON STANDARD DU BRUIT
Michel Fliess (LIX and INRIA Futurs)
Thanks to the nonstandard formalization of fast oscillating
functions, due to
P. Cartier and Y. Perrin, an appropriate mathematical framework is
derived for
new non-asymptotic estimation techniques, which do not necessitate any
statistical analysis of the noises corrupting any sensor. Various
applications
are deduced for multiplicative noises, for the length of the parametric
estimation windows, and for burst errors.
http://front.math.ucdavis.edu/cs.CE/0603003
---------------------------------------------------------------
4216. RECURRENCE OF RANDOM WALK TRACES
Itai Benjamini and Ori Gurel-Gurevich and Russell Lyons
We show that the edges crossed by a random walk in a network form a
recurrent
graph a.s. In fact, the same is true when those edges are weighted by
the
number of crossings.
http://front.math.ucdavis.edu/math.PR/0603060
---------------------------------------------------------------
4217. OPERATOR SCALING STABLE RANDOM FIELDS
Hermine Bierm\'{e} (MAP5) and Mark M. Meerschaert and Hans-Peter
Scheffler
A scalar valued random field is called operator-scaling if it
satisfies a
self-similarity property for some matrix E with positive real parts
of the
eigenvalues. We present a moving average and a harmonizable
representation of
stable operator scaling random fields by utilizing so called E-
homogeneous
functions. These fields also have stationary increments and are
stochastically
continuous. In the Gaussian case critical H\"{o}lder-exponents and the
Hausdorff-dimension of the sample paths are also obtained.
http://front.math.ucdavis.edu/math.PR/0602664
---------------------------------------------------------------
4218. PROCESSES ON UNIMODULAR RANDOM NETWORKS
David Aldous and Russell Lyons
We investigate unimodular random networks. Our motivations include their
characterization via reversibility of an associated random walk and
their
similarities to unimodular quasi-transitive graphs. We extend various
theorems
concerning random walks, percolation, spanning forests, and
amenability from
the known context of unimodular quasi-transitive graphs to the more
general
context of unimodular random networks. We give properties of a trace
associated
to unimodular random networks with applications to stochastic
comparison of
continuous-time random walk.
http://front.math.ucdavis.edu/math.PR/0603062
---------------------------------------------------------------
4219. MULTICRITICAL CONTINUOUS RANDOM TREES
J. Bouttier and P. Di Francesco and E. Guitter
We introduce generalizations of Aldous' Brownian Continuous Random
Tree as
scaling limits for multicritical models of discrete trees. These
discrete
models involve trees with fine-tuned vertex-dependent weights
ensuring a k-th
root singularity in their generating function. The scaling limit
involves
continuous trees with branching points of order up to k+1. We derive
explicit
integral representations for the average profile of this k-th order
multicritical continuous random tree, as well as for its history
distributions
measuring multi-point correlations. The latter distributions involve
non-positive universal weights at the branching points together with
fractional
derivative couplings. We prove universality by rederiving the same
results
within a purely continuous axiomatic approach based on the resolution
of a set
of consistency relations for the multi-point correlations. The
average profile
is shown to obey a fractional differential equation whose solution
involves
hypergeometric functions and matches the integral formula of the
discrete
approach.
http://front.math.ucdavis.edu/math-ph/0603007
---------------------------------------------------------------
4220. RANDOM ENERGY MODEL WITH COMPACT DISTRIBUTIONS
Nabin Kumar Jana
In this paper we study the Random energy model - so called toy model
of the
spin glass theory - where the underlying distributions are compactly
supported.
We prove a general theorem on the asymptotics of free energy and obtain
formulae in several interesting cases - like uniform distribution,
truncated
double exponential.
http://front.math.ucdavis.edu/math.PR/0602666
---------------------------------------------------------------
4221. THRESHOLD $THETA GEQ 2$ CONTACT PROCESSES ON HOMOGENEOUS TREES
Luiz Renato Fontes and Roberto H. Schonmann
We study the threshold $theta geq 2$ contact process on a homogeneous
tree
$T_b$ of degree $kappa = b + 1$, with infection parameter $lambda geq
0$ and
started from a product measure with density $p$. The corresponding
mean-field
model displays a discontinuous transition at a critical point
$lambda_c^{MF}(kappa,theta)$ and for $lambda geq lambda_c^{MF}
(kappa,theta)$ it
survives iff $p geq p_c^{MF}(kappa,theta,lambda)$, where this
critical density
satisfies $0 < p_c^{MF}(kappa,theta,lambda) < 1$, $lim_{lambda to infty}
p_c^{MF}(kappa,theta,lambda) = 0$. For large $b$, we show that the
process on
$T_b$ has a qualitatively similar behavior when $lambda$ is small,
including
the behavior at and close to the critical point $lambda_c(T_b,theta)
$. In
contrast, for large $lambda$ the behavior of the process on $T_b$ is
qualitatively distinct from that of the mean-field model in that the
critical
density has $p_c(T_b,theta,infty) := lim_{lambda to infty}
p_c(T_b,theta,lambda) > 0$. We also show that $lim_{b to infty} b
lambda_c(T_b,theta) = Phi_{theta}$, where $1 < Phi_2 < Phi_3 < ...$,
$lim_{theta to infty} Phi_{theta} = infty$, and $0 < liminf_{b to infty}
b^{theta(theta-1)} p_c(T_b,theta,infty) leq limsup_{b to infty}
b^{theta/(theta-1)} p_c(T_b,theta,infty) < infty$.
http://front.math.ucdavis.edu/math.PR/0603109
---------------------------------------------------------------
4222. STOCHASTIC EQUATION ON COMPACT GROUPS IN DISCRETE NEGATIVE TIME
Jir\^o Akahori and Chihiro Uenishi and Kouji Yano
In this paper a stochastic equation on compact groups in discrete
negative
time is studied. This is closely related to Tsirelson's stochastic
differential
equation, of which any solution is non-strong. How the group action
reflects on
the set of solutions is investigated. It is applied to generalize
Yor's result
and give a necessary and sufficient condition for existence of a strong
solution and for uniqueness in law.
http://front.math.ucdavis.edu/math.PR/0603113
---------------------------------------------------------------
4223. TRANSLATION-INVARIANCE OF TWO-DIMENSIONAL GIBBSIAN SYSTEMS OF
PARTICLES WITH INTERNAL DEGREES OF FREEDOM
Thomas Richthammer
The conservation of translation as a symmetry in two-dimensional
systems with
interaction is a classical subject of statistical mechanics. Here we
establish
such a result for Gibbsian systems of marked particles with two-body
interaction, where the interesting cases of singular, hard-core and
discontinuous interaction are included.
http://front.math.ucdavis.edu/math.PR/0603140
---------------------------------------------------------------
4224. REGULAR VARIATION AND SMILE ASYMPTOTICS
Shalom Benaim and Peter Friz
We consider risk-neutral returns and show how their tail asymptotics
translate directly to asymptotics of the implied volatility smile,
thereby
sharpening Roger Lee's celebrated moment formula. The theory of regular
variation provides the ideal mathematical framework to formulate and
prove such
results. The practical value of our formulae comes from the vast
literature on
tail asymptotics and our conditions are often seen to be true by simple
inspection of known results.
http://front.math.ucdavis.edu/math.PR/0603146
---------------------------------------------------------------
4225. SOME PARABOLIC PDES WHOSE DRIFT IS AN IRREGULAR RANDOM NOISE IN
SPACE
Francesco Russo (LAGA) and Gerald Trutnau (SFB 343)
We consider a new class of random partial differential equation of
parabolic
type where the stochastic term is constituted by an irregular noisy
drift, not
necessarily Gaussian. We provide a suitable interpretation and we study
existence. After freezing a realization of the drift (stochastic
process), we
study existence and uniqueness (in some suitable sense) of the
associated
parabolic equation and we investigate probabilistic interpretation.
http://front.math.ucdavis.edu/math.PR/0602669
---------------------------------------------------------------
4226. THE LEBESGUE DECOMPOSITION OF THE FREE ADDITIVE CONVOLUTION OF
TWO PROBABILITY DISTRIBUTIONS
Serban Teodor Belinschi
We prove that the free additive convolution of two Borel probability
measures
supported on the real line can have a component that is singular
continuous
with respect to the Lebesgue measure on the real line only if one of
the two
measures is a point mass. The density of the absolutely continuous
part with
respect to the Lebesgue measure is shown to be analytic wherever
positive and
finite. The atoms of the free additive convolution of Borel probability
measures on the real line have been described by Bercovici and
Voiculescu in a
previous paper.
http://front.math.ucdavis.edu/math.OA/0603104
---------------------------------------------------------------
4227. EXPONENTIAL RANDOM ENERGY MODEL
Nabin Kumar Jana
In this paper the Random Energy Model(REM) under exponential type
environment
is considered which includes double exponential and Gaussian cases.
Limiting
Free Energy is evaluated in these models. Limiting Gibbs'
distribution is
evaluated in the double exponential case.
http://front.math.ucdavis.edu/math.PR/0602670
---------------------------------------------------------------
4228. 2-FOLD AND 3-FOLD MIXING: WHY 3-DOT-TYPE COUNTEREXAMPLES ARE
IMPOSSIBLE IN ONE DIMENSION
Thierry De La Rue (LMRS)
V.A. Rohlin asked in 1949 whether 2-fold mixing implies 3-fold mixing
for a
stationary process indexed by Z, and the question remains open today.
In 1978,
F. Ledrappier exhibited a counterexample to the 2-fold mixing implies
3-fold
mixing problem, the so-called "3-dot system", but in the context of
stationary
random fields indexed by ZxZ. In this work, we first present an
attempt to
adapt Ledrappier's construction to the one-dimensional case, which
finally
leads to a stationary process which is 2-fold but not 3-fold mixing
conditionally to the sigma-algebra generated by some factor process.
Then,
using arguments coming from the theory of joinings, we will give some
strong
obstacles proving that Ledrappier's counterexample can not be fully
adapted to
one-dimensional stationary processes.
http://front.math.ucdavis.edu/math.PR/0603154
---------------------------------------------------------------
4229. LARGE DEVIATIONS AND LAWS OF THE ITERATED LOGARITHM FOR THE
LOCAL TIMES OF ADDITIVE STABLE PROCESSES
Xia Chen
We study the upper tail behaviors of the local times of the additive
stable
processes. Let $X_1(t),..., X_p(t)$ be independent, $d$-dimensional
symmetric
stable processes with stable index $0<\alpha\le 2$ and consider the
additive
stable process $\ol{X}(t_1,..., t_p)=X_1(t_1)+... +X_p(t_p)$. Under the
condition $d<\alpha p$, we obtain a precise form of large deviation
principle
for the local time $$
\eta^x\big([0,t]^p\big)=\int_0^t...\int_0^t\delta_x\big(X_1(s_1)+...
+X_p(s_p)\big)ds_1... ds_p $$ of the multi-parameter process $\ol{X}
(t_1,...,
t_p)$, and for its supremum norm
$\displaystyle\sup_{x\in\R^d}\eta^x\big([0,t]^p\big)$. Our results
apply to the
law of the iterated logarithm and our approach is based on Fourier
analysis,
moment computation and time exponentiation.
http://front.math.ucdavis.edu/math.PR/0603159
---------------------------------------------------------------
4230. A DILUTED VERSION OF THE PERCEPTRON MODEL
David Marquez-Carreras and Carles Rovira and Samy Tindel
This note is concerned with a diluted version of the perceptron
model. We
establish a replica symmetric formula at high temperature, which is
achieved by
studying the asymptotic behavior of a given spin magnetization. Our
main task
will be to identify the order parameter of the system.
http://front.math.ucdavis.edu/math.PR/0603162
---------------------------------------------------------------
4231. JOINT SINGULAR VALUE DISTRIBUTION OF TWO CORRELATED RECTANGULAR
GAUSSIAN MATRICES AND ITS APPLICATION
Shuangquan Wang and Ali Abdi
Let $\mathbf{H}=(h_{ij})$ and $\mathbf{G}=(g_{ij})$ be two $m\times n$,
$m\leq n$, random matrices, each with i.i.d complex zero-mean unit-
variance
Gaussian entries, with correlation between any two elements given by
$\mathbb{E}[h_{ij}g_{pq}^\star]=\rho \delta_{ip}\delta_{jq}$ such that
$|\rho|<1$, where ${}^\star$ denotes the complex conjugate and $
\delta_{ij}$ is
the Kronecker delta. Assume $\{s_k\}_{k=1}^m$ and $\{r_l\}_{l=1}^m$ are
unordered singular values of $\mathbf{H}$ and $\mathbf{G}$,
respectively, and
$s$ and $r$ are randomly selected from $\{s_k\}_{k=1}^m$ and $\{r_l\}_
{l=1}^m$,
respectively. In this paper, exact analytical closed-form expressions
are
derived for the joint probability distribution function (PDF) of
$\{s_k\}_{k=1}^m$ and $\{r_l\}_{l=1}^m$ using an Itzykson-Zuber-type
integral,
as well as the joint marginal PDF of $s$ and $r$, by a bi-orthogonal
polynomial
technique. These PDFs are of interest in multiple-input multiple-
output (MIMO)
wireless communication channels and systems.
http://front.math.ucdavis.edu/math.PR/0603170
---------------------------------------------------------------
4232. FREEZING TRANSITION OF THE DIRECTED POLYMER IN A $1+D$ RANDOM
MEDIUM : LOCATION OF THE CRITICAL TEMPERATURE AND UNUSUAL CRITICAL
PROPERTIES
Cecile Monthus and Thomas Garel
In dimension $d \geq 3$, the directed polymer in a random medium
undergoes a
phase transition between a free phase and a disorder dominated phase.
For the
latter, Fisher and Huse have proposed a droplet theory based on the
scaling of
the free energy fluctuations $\Delta F(l) \sim l^{\theta}$. On the
other hand,
in related growth models belonging to the KPZ universality class,
Forrest and
Tang have found that the height-height correlation function is
logarithmic at
the transition. For the directed polymer model at criticality, this
translates
into logarithmic free energy fluctuations $\Delta F_{T_c}(l) \sim (\ln
l)^{\sigma}$ with $\sigma=1/2$. In this paper, we propose a droplet
scaling
analysis exactly at criticality based on this logarithmic scaling.
Our main
conclusion is that the typical correlation length $\xi(T)$ of the low
temperature phase, diverges as $ \ln \xi(T) \sim (- \ln (T_c-T))^{1/
\sigma}
\sim (- \ln (T_c-T))^{2} $. Furthermore, the logarithmic dependence
of $\Delta
F_{T_c}(l)$ leads to the conclusion that the critical temperature $T_c$
actually coincides with the explicit upper bound $T_2$ derived by
Derrida and
coworkers, where $T_2$ corresponds to the temperature below which the
ratio
$\bar{Z_L^2}/(\bar{Z_L})^2$ diverges exponentially in $L$. Finally,
since the
Fisher-Huse droplet theory was initially introduced for the spin-
glass phase,
we briefly mention the similarities and differences with the directed
polymer
model. If one speculates that the free energy of droplet excitations for
spin-glasses is also logarithmic at $T_c$, one obtains a logarithmic
decay for
the mean square correlation function at criticality $\bar{C^2(r)}
\sim 1/(\ln r
)^{\sigma}$.
http://front.math.ucdavis.edu/cond-mat/0603041
---------------------------------------------------------------
4233. ON THE 2D ISING WULFF CRYSTAL NEAR CRITICALITY
Raphael Cerf and Reda Juerg Messikh
We study the behavior of the two-dimensional Ising model in a finite
box at
temperatures that are below, but very close to, the critical
temperature. In a
regime where the temperature approaches the critical point and,
simultaneously,
the size of the box grows fast enough, we establish a large deviation
principle
that proves the appearance of a round Wulff crystal
http://front.math.ucdavis.edu/math.PR/0603178
---------------------------------------------------------------
4234. GAME-THEORETIC VERSIONS OF STRONG LAW OF LARGE NUMBERS FOR
UNBOUNDED VARIABLES
Masayuki Kumon and Akimichi Takemura and Kei Takeuchi
We consider strong law of large numbers (SLLN) in the framework of
game-theoretic probability of Shafer and Vovk (2001). We prove
several versions
of SLLN for the case that Reality's moves are unbounded. Our game-
theoretic
versions of SLLN largely correspond to standard measure-theoretic
results.
However game-theoretic proofs are different from measure-theoretic
ones in the
explicit consideration of various hedges. In measure-theoretic proofs
existence
of moments are assumed, whereas in our game-theoretic proofs we assume
availability of various hedges to Skeptic for finite prices.
http://front.math.ucdavis.edu/math.PR/0603184
---------------------------------------------------------------
4235. ASYMPTOTICS FOR THE SMALL FRAGMENTS OF THE FRAGMENTATION AT NODES
Romain Abraham (MAPMO) and Jean-Fran\c{c}ois Delmas (CERMICS)
We consider the fragmentation at nodes of the L\'{e}vy continuous
random tree
introduced in a previous paper. In this framework we compute the
asymptotic for
the number of small fragments at time $\theta$. This limit is
increasing in
$\theta$ and discontinuous. In the $\alpha$-stable case the
fragmentation is
self-similar with index $1/\alpha$, with $\alpha \in (1,2)$ and the
results are
close to those Bertoin obtained for general self-similar
fragmentations but
with an additional assumtion which is not fulfilled here.
http://front.math.ucdavis.edu/math.PR/0603192
---------------------------------------------------------------
4236. FRAGMENTATION AT HEIGHT ASSOCIATED TO L\'{E}VY PROCESSES
Jean-Fran\c{c}ois Delmas (CERMICS)
We consider the height process of a L\'{e}vy process with no negative
jumps,
and its associated continuous tree representation. Using tools
developed by
Duquesne and Le Gall, we construct a fragmentation process at height,
which
generalizes the fragmentation at height of stable trees given by
Miermont. In
this more general framework, we recover that the dislocation measures
are the
same as the dislocation measures of the fragmentation at node
introduced by
Abraham and Delmas, up to a factor equal to the fragment size. We
also compute
the asymptotic for the number of small fragments.
http://front.math.ucdavis.edu/math.PR/0603193
---------------------------------------------------------------
4237. K-PROCESSES, SCALING LIMIT AND AGING FOR THE REM-LIKE TRAP MODEL
Luiz Renato Fontes and Pierre Mathieu
We study K-processes, which are Markov processes in a denumerable state
space, all of whose elements are stable, with the exception of a
single state,
starting from which the process enters finite sets of stable states with
uniform distribution. We show how these processes arise, in a particular
instance, as scaling limits of the REM-like trap model ``at low
temperature'',
and subsequently derive aging results for those models in this context.
http://front.math.ucdavis.edu/math.PR/0603198
---------------------------------------------------------------
4238. SELF-SIMILARITY AND FRACTIONAL BROWNIAN MOTIONS ON LIE GROUPS
F. Baudoin and L. Coutin
The goal of this paper is to define and study a notion of fractional
Brownian
motion on a Lie group. We define it as at the solution of a stochastic
differential equation driven by a linear fractional Brownian motion.
We show
that this process has stationary increments and satisfies a local
self-similar
property. Furthermore the Lie groups for which this self-similar
property is
global are characterized. Finally, we prove an integration by parts
formula on
the path group space and deduce the existence of a density.
http://front.math.ucdavis.edu/math.PR/0603199
---------------------------------------------------------------
4239. THE MAXIMUM OF A RANDOM WALK REFLECTED AT A GENERAL BARRIER
Niels Richard Hansen
We define the reflection of a random walk at a general barrier and
derive, in
case the increments are light tailed and have negative mean, a
necessary and
sufficient criterion for the global maximum of the reflected process
to be
finite a.s. If it is finite a.s., we show that the tail of the
distribution of
the global maximum decays exponentially fast and derive the precise
rate of
decay. Finally, we discuss an example from structural biology that
motivated
the interest in the reflection at a general barrier.
http://front.math.ucdavis.edu/math.PR/0603208
---------------------------------------------------------------
4240. ANALYSIS OF TOP TO BOTTOM-$K$ SHUFFLES
Sharad Goel
A deck of $n$ cards is shuffled by repeatedly moving the top card to
one of
the bottom $k_n$ positions uniformly at random. We give upper and
lower bounds
on the total variation mixing time for this shuffle as $k_n$ ranges
from a
constant to $n$. We also consider a symmetric variant of this shuffle
in which
at each step either the top card is randomly inserted into the bottom
$k_n$
positions or a random card from the bottom $k_n$ positions is moved
to the top.
For this reversible shuffle we derive bounds on the $L^2$ mixing
time. Finally,
we transfer mixing time estimates for the above shuffles to the lazy
top to
bottom-$k$ walks that move with probability 1/2 at each step.
http://front.math.ucdavis.edu/math.PR/0603209
---------------------------------------------------------------
4241. OVERSHOOTS AND UNDERSHOOTS OF L\'{E}VY PROCESSES
R. A. Doney and A. E. Kyprianou
We obtain a new fluctuation identity for a general L\'{e}vy process
giving a
quintuple law describing the time of first passage, the time of the last
maximum before first passage, the overshoot, the undershoot and the
undershoot
of the last maximum. With the help of this identity, we revisit the
results of
Kl\"{u}ppelberg, Kyprianou and Maller [Ann. Appl. Probab. 14 (2004)
1766--1801]
concerning asymptotic overshoot distribution of a particular class of
L\'{e}vy
processes with semi-heavy tails and refine some of their main
conclusions. In
particular, we explain how different types of first passage
contribute to the
form of the asymptotic overshoot distribution established in the
aforementioned
paper. Applications in insurance mathematics are noted with emphasis
on the
case that the underlying L\'{e}vy process is spectrally one sided.
http://front.math.ucdavis.edu/math.PR/0603210
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4242. ESTIMATION OF ANISOTROPIC GAUSSIAN FIELDS THROUGH RADON TRANSFORM
Hermine Bierm\'{e} (MAP5)
We estimate the anisotropic index of an anisotropic fractional Brownian
field. For all directions, we give a convergent estimator of the
value of the
anisotropic index in this direction, based on generalized quadratic
variations.
We also prove a central limit theorem. First we present a result of
identification based on the asymptotic behavior of the spectral
density of a
process. Then, we define Radon transforms of the anisotropic fractional
Brownian field and prove that these processes admit a spectral density
satisfying the previous assumptions.
http://front.math.ucdavis.edu/math.ST/0602663
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4243. ON A NONHIERARCHICAL VERSION OF THE GENERALIZED RANDOM ENERGY
MODEL
Erwin Bolthausen and Nicola Kistler
We introduce a natural nonhierarchical version of Derrida's generalized
random energy model. We prove that, in the thermodynamical limit, the
free
energy is the same as that of a suitably constructed GREM.
http://front.math.ucdavis.edu/math.PR/0603212
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4244. A SCHEME FOR SIMULATING ONE-DIMENSIONAL DIFFUSION PROCESSES
WITH DISCONTINUOUS COEFFICIENTS
Antoine Lejay and Miguel Martinez
The aim of this article is to provide a scheme for simulating diffusion
processes evolving in one-dimensional discontinuous media. This
scheme does not
rely on smoothing the coefficients that appear in the infinitesimal
generator
of the diffusion processes, but uses instead an exact description of the
behavior of their trajectories when they reach the points of
discontinuity.
This description is supplied with the local comparison of the
trajectories of
the diffusion processes with those of a skew Brownian motion.
http://front.math.ucdavis.edu/math.PR/0603214
---------------------------------------------------------------
4245. STOCHASTIC DYNAMICS OF DISCRETE CURVES AND EXCLUSION PROCESSES.
PART 1: HYDRODYNAMIC LIMIT OF THE ASEP SYSTEM
Guy Fayolle and Cyril Furtlehner
This report is the foreword of a series dedicated to stochastic
deformations
of curves. Problems are set in terms of exclusion processes, the
ultimate goal
being to derive hydrodynamic limits for these systems after proper
scalings. In
this study, solely the basic \textsc{asep} system on the torus is
analyzed. The
usual sequence of empirical measures, converges in probability to a
deterministic measure, which is the unique weak solution of a Cauchy
problem.
The method presents some new features, letting hope for extensions to
higher
dimension. It relies on the analysis of a family of parabolic
differential
operators, involving variational calculus. Namely, the variables are
the values
of functions at given points, their number being possibly infinite.
http://front.math.ucdavis.edu/math.PR/0603215
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4246. CONTINUITY FOR SELF-DESTRUCTIVE PERCOLATION IN THE PLANE
J. van den Berg and R. Brouwer and B. Vagvolgyi
A few years ago two of us introduced, motivated by the study of certain
forest-fireprocesses, the self-destructive percolation model
(abbreviated as
sdp model). A typical configuration for the sdp model with parameters
p and
delta is generated in three steps: First we generate a typical
configuration
for the ordinary percolation model with parameter p. Next, we make
all sites in
the infinite occupied cluster vacant. Finally, each site that was
already
vacant in the beginning or made vacant by the above action, becomes
occupied
with probability delta (independent of the other sites).
Let theta(p, delta) be the probability that some specified vertex
belongs, in
the final configuration, to an infinite occupied cluster. In our
earlier paper
we stated the conjecture that, for the square lattice and other planar
lattices, the function theta has a discontinuity at points of the
form (p_c,
delta), with delta sufficiently small. We also showed remarkable
consequences
for the forest-fire models.
The conjecture naturally raises the question whether the function
theta is
continuous outside some region of the above mentioned form. We prove
that this
is indeed the case. An important ingredient in our proof is a (somewhat
stronger form of a) recent ingenious RSW-like percolation result of
Bollob\'{a}s and Riordan.
http://front.math.ucdavis.edu/math.PR/0603223
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4247. ELEMENTS OF STOCHASTIC CALCULUS VIA REGULARISATION
Francesco Russo and Pierre Vallois
This paper first summarizes the foundations of stochastic calculus via
regularization and constructs through this procedure It\^o and
Stratonovich
integrals. In the second part, a survey and new results are presented in
relation with finite quadratic variation processes, Dirichlet and weak
Dirichlet processes.
http://front.math.ucdavis.edu/math.PR/0603224
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4248. ON THE CRITICAL BEHAVIOR AT THE LOWER PHASE TRANSITION OF THE
CONTACT PROCESS
Michael Aizenman and Paul Jung
We present general results for the contact process by a method which
applies
to all transitive graphs of bounded degree, including graphs of
exponential
growth. The model's infection rates are varied though a common control
parameter, for which two natural transition points are defined as: i.
$\lambda_T$, the value up to which the infection dies out
exponentially fast if
introduced at a single site, and ii. $\lambda_H$, the threshold for the
existence of an invariant measure with a non-vanishing density of
infected
sites. It is shown here that for all transitive graphs the two
thresholds
coincide. The method, which proceeds through partial differential
inequalities
for the infection density, yields also generally valid bounds on two
related
critical exponents. The work extends existing results whose
derivations were
restricted to either the discrete-time versions of the contact
process or to
graphs with subexponential growth.
http://front.math.ucdavis.edu/math.PR/0603227
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4249. STRONG LOCALIZATION AND MACROSCOPIC ATOMS FOR DIRECTED POLYMERS
Vincent Vargas (PMA)
In this article, we derive strong localization results for directed
polymers
in random environment. We show that at "low temperature" the polymer
measure is
asymptotically concentrated at a few points of macroscopic mass (we
call these
points epsilon-atoms). These results are derived assuming weak
conditions on
the tail decay of the random environment.
http://front.math.ucdavis.edu/math.PR/0603233
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4250. AN INVARIANCE PRINCIPLE FOR NEW WEAKLY DEPENDENT STATIONARY
MODELS USING SHARP MOMENT ASSUMPTIONS
Paul Doukhan (LS-CREST and SAMOS) and Olivier Wintenberger (SAMOS)
This paper is aimed at sharpen a weak invariance principle for
stationary
sequences in Doukhan & Louhichi (1999). Our assumption is both beyond
mixing
and the causal $\theta$-weak dependence in Dedecker and Doukhan
(2003); those
authors obtained a sharp result which improves on an optimal one in
Doukhan
{\it et alii} (1995) under strong mixing. We prove this result and we
also
precise convergence rates under existence of moments with order $>2$
while
Doukhan & Louhichi (1999) assume a moment of order $>4$. Analogously
to those
authors, we use a non-causal condition to deal with some general
classes of
stationary and weakly dependent sequences. Besides the previously
used $\eta$-
and $\kappa$-weak dependence conditions, we introduce a mixed condition,
$\lambda$, adapted to consider Bernoulli shifts with dependent inputs.
http://front.math.ucdavis.edu/math.ST/0603221
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4251. THRESHOLDS AND EXPECTATION THRESHOLDS
Jeff Kahn and Gil Kalai
Consider a random graph G in G(n,p) and the graph property: G
contains a copy
of a specific graph H. (An example to keep in mind: H is a
Hamiltonian cycle.)
Let p be the minimal value for which the expected number of copies of
H' in G
is at least 1/2 for every subgraph H' of H. Let q be the value for
which the
probability that G contains a copy of H is 1/2. Conjecture: q/p = O
(log n).
Related conjectures for general Boolean functions, and a possible
connection
with discrete isoperimetry are discussed.
http://front.math.ucdavis.edu/math.CO/0603218
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4252. LARGE N LIMIT OF GAUSSIAN RANDOM MATRICES WITH EXTERNAL SOURCE,
PART
Pavel M. Bleher and Arno B.J. Kuijlaars
We consider the double scaling limit in the random matrix ensemble
with an
external source $\frac{1}{Z_n} e^{-n \Tr({1/2}M^2 -AM)} dM$ defined
on $n\times
n$ Hermitian matrices, where $A$ is a diagonal matrix with two
eigenvalues $\pm
a$ of equal multiplicities. The value $a=1$ is critical since the
eigenvalues
of $M$ accumulate as $n \to \infty$ on two intervals for $a > 1$ and
on one
interval for $0 < a < 1$. These two cases were treated in Parts I and
II, where
we showed that the local eigenvalue correlations have the universal
limiting
behavior known from unitary random matrix ensembles. For the critical
case
$a=1$ new limiting behavior occurs which is described in terms of
Pearcey
integrals, as shown by Br\'ezin and Hikami, and Tracy and Widom. We
establish
this result by applying the Deift/Zhou steepest descent method to a
$3 \times
3$-matrix valued Riemann-Hilbert problem which involves the
construction of a
local parametrix out of Pearcey integrals. We resolve the main
technical issue
of matching the local Pearcey parametrix with a global outside
parametrix by
modifying an underlying Riemann surface.
http://front.math.ucdavis.edu/math-ph/0602064
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4253. A FORWARD--BACKWARD STOCHASTIC ALGORITHM FOR QUASI-LINEAR PDES
Fran\c{c}ois Delarue and St\'{e}phane Menozzi
We propose a time-space discretization scheme for quasi-linear parabolic
PDEs. The algorithm relies on the theory of fully coupled forward--
backward
SDEs, which provides an efficient probabilistic representation of
this type of
equation. The derivated algorithm holds for strong solutions defined
on any
interval of arbitrary length. As a bypass product, we obtain a
discretization
procedure for the underlying FBSDE. In particular, our work provides an
alternative to the method described in [Douglas, Ma and Protter
(1996) Ann.
Appl. Probab. 6 940--968] and weakens the regularity assumptions
required in
this reference.
http://front.math.ucdavis.edu/math.PR/0603250
---------------------------------------------------------------
4254. A NEW INVERSE FORMULA FOR THE LAPLAS'S TRANSFORMATION
Andrey Pavlov
In the article is proved,that the complex part of the analytical
continuation
of the LL(Z(x)) on the negative axis is equal to cZ(x),c=const., were
Z(x) is
the odd function from the wide class of functions,L(Z(x)) is the
transformation
of Laplas.
http://front.math.ucdavis.edu/math.PR/0603258
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