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Probability Abstracts 91
This document contains abstracts 3954-4109 from
Jan-1-2006 to Feb-28-2006.
They have been mailed on March 1, 2006.
Author(s): Laurent Decreusefond and Pascal Moyal
Abstract: In this paper we present the fluid limit of an heavily loaded Earliest
Deadline First queue with impatient customers, represented by a measure-valued
process keeping track of residual time-credits of lost and waiting customers.
This fluid limit is the solution of an integrated transport equation. We then
use this fluid limit to derive fluid approximations of the processes counting
the number of waiting and already lost customers.
http://arXiv.org/abs/math/0512660
http://front.math.ucdavis.edu/math.PR/0512660
(alternate)
Author(s): Eitan Bachmat
Abstract: We consider the problem of estimating the average tour length of the
asymmetric TSP arising from the disk scheduling problem with a linear seek
function and a probability distribution on the location of I/O requests. The
optimal disk scheduling algorithm of Andrews, Bender and Zhang is interpreted
as a simple peeling process on points in a 2 dimensional space-time w.r.t the
causal structure. The patience sorting algorithm for finding the longest
increasing subsequence in a permutation can be given a similar interpretation.
Using this interpretation we show that the optimal tour length is the length of
the maximal curve with respect to a Lorentzian metric on the surface of the
disk drive. This length can be computed explicitly in some interesting cases.
When the probability distribution is assumed uniform we provide finer
asymptotics for the tour length. The interpretation also provides a better
understanding of patience sorting and allows us to extend a result of Aldous
and Diaconis on pile sizes
http://arXiv.org/abs/math/0601025
http://front.math.ucdavis.edu/math.OC/0601025
(alternate)
Author(s): Jean-Maxime Labarbe (LM-Versailles) and Jean-Fran\c{c}ois Marckert (LaBRI)
Abstract: A Bernoulli random walk is a random trajectory starting from 0 and having
i.i.d. increments, each of them being $+1$ or -1, equally likely. The other
families cited in the title are Bernoulli random walks under various
conditionings. A peak in a trajectory is a local maximum. In this paper, we
condition the families of trajectories to have a given number of peaks. We show
that, asymptotically, the main effect of setting the number of peaks is to
change the order of magnitude of the trajectories. The counting process of the
peaks, that encodes the repartition of the peaks in the trajectories, is also
studied. It is shown that suitably normalized, it converges to a Brownian
bridge which is independent of the limiting trajectory. Applications in terms
of plane trees and parallelogram polyominoes are also provided.
http://arXiv.org/abs/math/0601624
http://front.math.ucdavis.edu/math.PR/0601624
(alternate)
Author(s): Giada Basile (CEREMADE) and Cedric Bernardin (UMPA-ENSL) and Stefano Olla (CEREMADE)
Abstract: We present here complete mathematical proofs of the results announced in
cond-mat/0509688. We introduce a model whose thermal conductivity diverges in
dimension 1 and 2, while it remains finite in dimension 3. We consider a system
of harmonic oscillators perturbed by a stochastic dynamics conserving momentum
and energy. We compute the nite-size thermal conductivity via Green-Kubo
formula. In the limit as the size N of the system goes to in nity, conductivity
diverges like N in dimension 1 and like lnN in dimension 2. Conductivity
remains finite if dimesion is 3 or higher or if a pinning (on site potential)
is present.
http://arXiv.org/abs/cond-mat/0601544
http://front.math.ucdavis.edu/cond-mat/0601544
(alternate)
Author(s): Alex V. Kontorovich and Yakov G. Sinai
Abstract: The (3x+1)-Map, T, acts on the set, Pi, of positive integers not divisible by
2 or 3. It is defined by T(x) = (3x+1)/2^k, where k is the largest integer for
which T(x) is an integer. The (3x+1)-Conjecture asks if for every x in Pi there
exists an integer, n, such that T^n (x) = 1. The Statistical (3x+1)-Conjecture
asks the same question, except for a subset of Pi of density 1. The Structure
Theorem proven in \cite{sinai} shows that infinity is in a sense a repelling
point, giving some reasons to expect that the (3x+1)-Conjecture may be true. In
this paper, we present the analogous theorem for some generalizations of the
(3x+1)-Map, and expand on the consequences derived in \cite{sinai}. The
generalizations we consider are determined by positive coprime integers, d and
g, with g > d >= 2, and a periodic function, h(x). The map T is defined by the
formula T(x) = (gx+h(gx))/d^k, where k is again the largest integer for which
T(x) is an integer. We prove an analogous Structure Theorem for (d,g,h)-Maps,
and that the probability distribution corresponding to the density converges to
the Wiener measure with the drift log(g) - d/(d-1)log(d) and positive diffusion
constant. This shows that it is natural to expect that typical trajectores
return to the origin if log(g) - d/(d-1) log(d) <0 and escape to infinity
otherwise.
http://arXiv.org/abs/math/0601622
http://front.math.ucdavis.edu/math.NT/0601622
(alternate)
Author(s): Soumik Pal
Abstract: Consider an agent who enters a financial market on day t = 0 with an initial
capital amount x. He invests this amount on stocks and the money market, and by
day t = T, has generated a wealth W . He is given a convex class of probability
measures (called scenarios) and a real-valued function (or floors)
corresponding to each scenario. The agent faces the constraints that the
expectation of W under each scenario must not be less than the corresponding
floor. We call x acceptable if one can start with x and successfully generate W
satisfying these constraints. The set of acceptable x is a half-line in R,
unbounded from above. We show that under some regularity conditions on the set
of scenarios and the floor function, the infimum of this set is given by the
supremum of the floors over all scenarios under which S is a martingale.
http://arXiv.org/abs/math/0601627
http://front.math.ucdavis.edu/math.PR/0601627
(alternate)
Author(s): Yaozhong Hu and David Nualart
Abstract: We derive estimates for the solutions to differential equations driven by a
H\"older continuous function of order $\beta>1/2$. As an application we deduce
the existence of moments for the solutions to stochastic partial differential
equations driven by a fractional Brownian motion with Hurst parameter
$H>{1/2}$.
http://arXiv.org/abs/math/0601628
http://front.math.ucdavis.edu/math.PR/0601628
(alternate)
Author(s): Tom Carroll and Joaquim Ortega-Cerd\`a
Abstract: We consider a collection of balls in Euclidean space and the problem of
determining if Brownian motion has a positive probability of avoiding all the
balls
http://arXiv.org/abs/math/0601632
http://front.math.ucdavis.edu/math.PR/0601632
(alternate)
Author(s): Nicolas Champagnat (WIAS) and Amaury Lambert (FESE)
Abstract: The biological theory of adaptive dynamics proposes a description of the
long-time evolution of an asexual population, based on the assumptions of large
population, rare mutations and small mutation steps, that lead to a
deterministic ODE, called 'canonical equation of adaptive dynamics'. However,
in order to include the effect of genetic drift in this description, we have to
apply a limit of weak selection to a finite stochastically fluctuating discrete
population subject to competition in the logistic branching fashion. We start
with the study of the particular case of two competing subpopulations resident
and mutant) and seek explicit first-order formulae for the probability of
fixation of the mutant, also interpreted as the mutant's fitness, in the
vicinity of neutrality. In particular, the first-order term is a linear
combination of products of functions of the initial mutant frequency times
functions of the initial total population size, called invasibility
coefficients (fertility, defence, aggressiveness, isolation, survival). Then we
apply a limit of rare mutations to a population subject to mutation, birth and
competition where the number of coexisting types may fluctuate, while keeping
the population size finite. This leads to a jump process, the so-called 'trait
substitution sequence', where evolution proceeds by successive invasions and
fixations of mutant types. Finally, we apply a limit of weak selection (small
mutation steps) to this jump process, that leads to a diffusion process of
evolution, called 'canonical diffusion of adaptive dynamics', in which genetic
drift is combined with directional selection driven by the fitness gradient.
http://arXiv.org/abs/math/0601643
http://front.math.ucdavis.edu/math.PR/0601643
(alternate)
Author(s): Tryphon T. Georgiou
Abstract: The maximum entropy ansatz, as it is often invoked in the context of
time-series analysis, suggests the selection of a power spectrum which is
consistent with autocorrelation data and corresponds to a random process least
predictable from past observations. We introduce and compare a class of spectra
with the property that the underlying random process is least predictable at
any given point from the complete set of past and future observations. In this
context, randomness is quantified by the size of the corresponding smoothing
error and deterministic processes are characterized by integrability of the
inverse of their power spectral densities--as opposed to the log-integrability
in the classical setting. The power spectrum which is consistent with a partial
autocorrelation sequence and corresponds to the most random process in this new
sense, is no longer rational but generated by finitely many fractional-poles.
http://arXiv.org/abs/math/0601648
http://front.math.ucdavis.edu/math.PR/0601648
(alternate)
Author(s): Soumik Pal
Abstract: Let X be a random variable. We shall call an independent random variable Y to
be a symmetrizer for X, if X+Y is symmetric around zero. A random variable is
said to be symmetry resistant if the variance of any symmetrizer Y, is never
smaller than the variance of X itself. We prove that a Bernoulli(p) random
variable is symmetry resistant if and only if p is not 1/2. This is an old
problem proved in 1999 by Kagan, Mallows, Shepp, Vanderbei & Vardi using linear
programming principles. We reprove it here using completely probabilistic tools
using Skorokhod embedding and Ito's rule.
http://arXiv.org/abs/math/0601652
http://front.math.ucdavis.edu/math.PR/0601652
(alternate)
Author(s): Noam Berger
Abstract: We show that Random Walk in uniformly elliptic i.i.d. environment in
dimension 5 and higher has at most one non-zero limiting velocity. In
particular this proves a law of large numbers in the distributionally symmetric
case and establishes connections between different conjectures.
http://arXiv.org/abs/math/0601656
http://front.math.ucdavis.edu/math.PR/0601656
(alternate)
Author(s): Julien Berestycki and Nathanael Berestycki and Jason Schweinsberg
Abstract: For a finite measure $\Lambda$ on $[0,1]$, the $\Lambda$-coalescent is a
coalescent process such that, whenever there are $b$ clusters, each $k$-tuple
of clusters merges into one at rate $\int_0^1 x^{k-2} (1-x)^{b-k} \Lambda(dx)$.
It has recently been shown that if $1 < \alpha < 2$, the $\Lambda$-coalescent
in which $\Lambda$ is the Beta$(2-\alpha, \alpha)$ distribution can be used to
describe the genealogy of a continuous-state branching process (CSBP) with an
$\alpha$-stable branching mechanism. Here we use facts about CSBPs to establish
new results about the small-time asymptotics of beta coalescents. We prove an
a.s. limit theorem for the number of blocks at small times, and we establish
results about the sizes of the blocks. We also calculate the Hausdorff and
packing dimensions of a metric space associated with the beta coalescents, and
we find the sum of the lengths of the branches in the coalescent tree, both of
which are determined by the behavior of coalescents at small times. We extend
most of these results to other $\Lambda$-coalescents for which $\Lambda$ has
the same asymptotic behavior near zero as the Beta$(2-\alpha, \alpha)$
distribution. This work complements recent work of Bertoin and Le Gall, who
also used CSBPs to study small-time properties of $\Lambda$-coalescents.
http://arXiv.org/abs/math/0601032
http://front.math.ucdavis.edu/math.PR/0601032
(alternate)
Author(s): Jean Bertoin (PMA)
Abstract: We consider a Langevin process with white noise random forcing. We suppose
that the energy of the particle is instantaneously absorbed when it hits some
fixed obstacle. We show that nonetheless, the particle can be instantaneously
reflected, and study some properties of this reflecting solution.
http://arXiv.org/abs/math/0601657
http://front.math.ucdavis.edu/math.PR/0601657
(alternate)
Author(s): Philippe Carmona (LMJL) and Yueyun Hu (LAGA)
Abstract: In this note we show that in any dimension $d$, the strong disorder property
implies the strong localization property. This is established for a continuous
time model of directed polymers in a random environment : the parabolic
Anderson Model.
http://arXiv.org/abs/math/0601670
http://front.math.ucdavis.edu/math.PR/0601670
(alternate)
Author(s): Boris Tsirelson
Abstract: A random dense countable set is characterized (in distribution) by
independence and stationarity. Two examples are `Brownian local minima' and
`unordered infinite sample'. They are identically distributed. A framework for
such concepts, proposed here, includes a wide class of random equivalence
classes.
http://arXiv.org/abs/math/0601673
http://front.math.ucdavis.edu/math.PR/0601673
(alternate)
Author(s): Milos Stojakovic and Tibor Szabo
Abstract: We introduce and study Maker/Breaker-type positional games on random graphs.
Our main concern is to determine the threshold probability $p_{F}$ for the
existence of Maker's strategy to claim a member of $F$ in the unbiased game
played on the edges of random graph $G(n,p)$, for various target families $F$
of winning sets. More generally, for each probability above this threshold we
study the smallest bias $b$ such that Maker wins the $(1\:b)$ biased game. We
investigate these functions for a number of basic games, like the connectivity
game, the perfect matching game, the clique game and the Hamiltonian cycle
game.
http://arXiv.org/abs/math/0601659
http://front.math.ucdavis.edu/math.CO/0601659
(alternate)
Author(s): Claude Dellacherie and Servet Martinez and Jaime San Martin
Abstract: We prove that the class of GUM matrices is the largest class of bi-potential
matrices stable under Hadamard increasing functions. We also show that any
power greater than 1, in the sense of Hadamard functions, of an inverse
M-matrix is also inverse M-matrix showing a conjecture stated in Neumann 1998.
We study the class of filtered matrices, which include naturally the GUM
matrices, and present some sufficient conditions for a filtered matrix to be a
bi-potential.
http://arXiv.org/abs/math/0601688
http://front.math.ucdavis.edu/math.PR/0601688
(alternate)
Author(s): Shige Peng
Abstract: We develop a notion of nonlinear expectation --G-expectation-- generated by a
nonlinear heat equation with infinitesimal generator G. We first study
multi-dimensional G-normal distributions. With this nonlinear distribution we
can introduce our G-expectation under which the canonical process is a multi
dimensional G-Brownian motion. We then establish the related stochastic
calculus, especially stochastic integrals of Ito's type with respect to our
G-Brownian motion and derive the related Ito's formula. We have also obtained
the existence and uniqueness of stochastic differential equation under our
G-expectation.
http://arXiv.org/abs/math/0601699
http://front.math.ucdavis.edu/math.PR/0601699
(alternate)
Author(s): Luc Bouten and Ramon van Handel and Matthew James
Abstract: This paper provides an introduction to quantum filtering theory. An
introduction to quantum probability theory is given, focusing on the spectral
theorem and the conditional expectation as a least squares estimate, and
culminating in the construction of Wiener and Poisson processes on the Fock
space. We describe the quantum It\^o calculus and its use in the modelling of
physical systems. We use both reference probability and innovations methods to
obtain quantum filtering equations for system-probe models from quantum optics.
http://arXiv.org/abs/math/0601741
http://front.math.ucdavis.edu/math.OC/0601741
(alternate)
Author(s): Shige Peng
Abstract: We introduce a notion of nonlinear expectation --$G$--expectation-- generated
by a nonlinear heat equation with infinitesimal generator $G$. We first discuss
the notion of $G$--standard normal distribution. With this nonlinear
distribution we can introduce our $G$--expectation under which the canonical
process is a $G$--Brownian motion. We then establish the related stochastic
calculus, especially stochastic integrals of It\^{o}'s type with respect to our
$G$--Brownian motion and derive the related It\^{o}'s formula. We have also
give the existence and uniqueness of stochastic differential equation under our
$G$--expectation. As compared with our previous framework of $g$--expectations,
the theory of $G$--expectation is intrinsic in the sense that it is not based
on a given (linear) probability space.
http://arXiv.org/abs/math/0601035
http://front.math.ucdavis.edu/math.PR/0601035
(alternate)
Author(s): Ilya Pavlyukevich
Abstract: We consider a dynamical system in R driven by a vector field -U', where U is
a multi-well potential satisfying some regularity conditions. We perturb this
dynamical system by a Levy noise of small intensity and such that the heaviest
tail of its Levy measure is regularly varying. We show that the perturbed
dynamical system exhibits metastable behaviour i.e. on a proper time scale it
reminds of a Markov jump process taking values in the local minima of the
potential U. Due to the heavy-tail nature of the random perturbation, the
results differ strongly from the well studied purely Gaussian case.
http://arXiv.org/abs/math/0601771
http://front.math.ucdavis.edu/math.PR/0601771
(alternate)
Author(s): Harald Luschgy (PMA) and Gilles Pag\`{e}s (PMA)
Abstract: We investigate the connections between the mean pathwise regularity of
stochastic processes and their $L^r(\P$)-functional quantization rate as random
variables taking values in some $L^p([0,T],dt)$-spaces ($<0p\le r$). Our main
tool is the Haar basis. We then emphasize that the derived functional
quantization rate may be optimal (like for the Brownian motion) or not (like
for the Poisson process). Then, we focus on the specific family of L\'evy
processes for which we derive a general quantization rate based on the regular
variation properties of its L\'evy measure at 0. The case of compound Poisson
processes which appears as degenerate in the former approach, are studied
specifically: one observes some rates which are in-between finite dimensional
and infinite dimensional "usual" rates.
http://arXiv.org/abs/math/0601774
http://front.math.ucdavis.edu/math.PR/0601774
(alternate)
Author(s): P. Salminen and O. Wallin
Abstract: In this paper we study perpetual integral functionals of diffusions. Our
interest is focused on cases where such functionals can be expressed as first
hitting times for some other diffusions. In particular, we generalize the
result which connects one-sided functionals of Brownian motion with drift with
first hitting times of reflecting diffusions.
Interpretating perpetual integral functionals as hitting times allows us to
compute numerically their distributions by applying numerical algorithms for
hitting times. Hereby, we discuss two approaches: the numerical inversion of
the Laplace transform of the first hitting time and the numerical solution of
the PDE associated with the distribution function of the first hitting time.
For numerical inversion of Laplace tranforms we have implemented the Euler
algorithm developed by Abate and Whitt. However, perpetuities lead often to
diffusions for which the explicit forms of the Laplace transforms of first
hitting times are not available. In such cases, and also otherwise, algorithms
for numerical solutions of PDE's can be evoked. In particular, we analyze the
Kolmogorov PDE of some diffusions appearing in our work via the Crank-Nicolson
scheme.
http://arXiv.org/abs/math/0601775
http://front.math.ucdavis.edu/math.PR/0601775
(alternate)
Author(s): Rohit Deo (IOMS) and Clifford M. Hurvich (IOMS) and Philippe Soulier (MODAL'X), Yi Wang (IOMS)
Abstract: We establish sufficient conditions on durations that are stationary with
finite variance and memory parameter $d \in [0,1/2)$ to ensure that the
corresponding counting process $N(t)$ satisfies $\textmd{Var} N(t) \sim C
t^{2d+1}$ ($C>0$) as $t \to \infty$, with the same memory parameter $d \in
[0,1/2)$ that was assumed for the durations. Thus, these conditions ensure that
the memory in durations propagates to the same memory parameter in counts and
therefore in realized volatility. We then show that any utoregressive
Conditional Duration ACD(1,1) model with a sufficient number of finite moments
yields short memory in counts, while any Long Memory Stochastic Duration model
with $d>0$ and all finite moments yields long memory in counts, with the same
$d$.
http://arXiv.org/abs/math/0601742
http://front.math.ucdavis.edu/math.ST/0601742
(alternate)
Author(s): Richard F. Bass and Xia Chen and and Jay Rosen
Abstract: Given a symmetric random walk in $Z^2$ with finite second moments, let $R_n$
be the range of the random walk up to time $n$. We study moderate deviations
for $R_n -E R_n$ and $E R_n -R_n$. We also derive the corresponding laws of the
iterated logarithm.
http://arXiv.org/abs/math/0602001
http://front.math.ucdavis.edu/math.PR/0602001
(alternate)
Author(s): Benjamin Hoff
Abstract: We introduce the (path-valued) Brownian frame process whose evaluation at
time t is the sample path of the underlying Brownian motion run from time t-1
to t. Due to its connections with Gaussian Volterra processes and SDDEs this is
an interesting object to study. The first part deals with path-wise properties
of the Brownian frame process in the p-variation norm. The second part shows
the non-existence of a Levy area random variable in a particular norm,
revealing the difficulty in establishing a Rough Path integration theory for
the Brownian Frame process.
http://arXiv.org/abs/math/0602008
http://front.math.ucdavis.edu/math.PR/0602008
(alternate)
Author(s): Rui Vilela Mendes
Abstract: Based on criteria of mathematical simplicity and consistency with empirical
market data, a stochastic volatility model is constructed, the volatility
process being driven by fractional noise. Price return statistics and
asymptotic behavior are derived from the model and compared with data.
http://arXiv.org/abs/math/0602013
http://front.math.ucdavis.edu/math.PR/0602013
(alternate)
Author(s): Randal Douc (CMAP) and Arnaud Guillin (CEREMADE) and Eric Moulines (LTCI)
Abstract: This paper studies limit theorems for Markov Chains with general state space
under conditions which imply subgeometric ergodicity. We obtain a central limit
theorem and moderate deviation principles for additive not necessarily bounded
functional of the Markov chains under drift and minorization conditions which
are weaker than the Foster-Lyapunov conditions. The regeneration-split chain
method and a precise control of the modulated moment of the hitting time to
small sets are employed in the proof.
http://arXiv.org/abs/math/0601036
http://front.math.ucdavis.edu/math.PR/0601036
(alternate)
Author(s): Shmuel Friedland and Elliot Krop
Abstract: We give simple necessary and sufficient conditions for the
inclusion-exclusion identity to hold for an infinite countable number of sets.
In terms of a random variable, whose range are nonnegative integers, this
condition is equivalent to the convergence to zero of binomial moments. Some
standard extensions of the countable inclusion-exclusion identity are also
given.
http://arXiv.org/abs/math/0602035
http://front.math.ucdavis.edu/math.PR/0602035
(alternate)
Author(s): Thomas Mountford and Leandro P. R. Pimentel and Glauco Valle
Abstract: We study a class of nearest-neighbor discrete time integer random walks
introduced by Zerner, the so called multi-excited random walks. The jump
probabilities for such random walker have a drift to the right whose intensity
depends on a random or non-random environment that also evolves in time
according to the last visited site. A complete description of the recurrence
and transience phases was given by Zerner under fairly general assumptions for
the environment. We contribute in this paper with some results that allows us
to point out if the random walker speed is strictly positive or not in the
transient case for a class of non-random environments.
http://arXiv.org/abs/math/0602041
http://front.math.ucdavis.edu/math.PR/0602041
(alternate)
Author(s): Yaozhong Hu and David Nualart
Abstract: Using fractional calculus we define integrals of the form $%
\int_{a}^{b}f(x_{t})dy_{t}$, where $x$ and $y$ are vector-valued H\"{o}lder
continuous functions of order $\displaystyle \beta \in (\frac13, \frac12)$ and
$f$ is a continuously differentiable function such that $f'$ is
$\lambda$-H\"oldr continuous for some $\lambda>\frac1\beta-2$. Under some
further smooth conditions on $f$ the integral is a continuous functional of
$x$, $y$, and the tensor product $x\otimes y$ with respect to the H\"{o}lder
norms. We derive some estimates for these integrals and we solve differential
equations driven by the function $y$. We discuss some applications to
stochastic integrals and stochastic differential equations.
http://arXiv.org/abs/math/0602050
http://front.math.ucdavis.edu/math.PR/0602050
(alternate)
Author(s): Cristian Giardina' and Shannon Starr
Abstract: We compute the pressure of the random energy model (REM) and generalized
random energy model(GREM) by establishing variational upper and lower bounds.
For the upper bound, we generalize Guerra's ``broken replica symmetry
bounds",and identify the random probability cascade as the appropriate random
overlap structure for the model. For the REM the lower bound is obtained, in
the high temperature regime using Talagrand's concentration of measure
inequality, and in the low temperature regime using convexity and the high
temperature formula. The lower bound for the GREM follows from the lower bound
for the REM by induction. While the argument for the lower bound is fairly
standard, our proof of the upper bound is new.
http://arXiv.org/abs/math-ph/0601068
http://front.math.ucdavis.edu/math-ph/0601068
(alternate)
Author(s): Terence Tao
Abstract: We introduce a correspondence principle (analogous to the Furstenberg
correspondence principle) that allows one to extract an infinite random graph
or hypergraph from a sequence of increasingly large deterministic graphs or
hypergraphs. As an application we present a new (infinitary) proof of the
hypergraph removal lemma of Nagle-Schacht-R\"odl-Skokan and Gowers, which does
not require the hypergraph regularity lemma and requires significantly less
computation. This in turn gives new proofs of several corollaries of the
hypergraph removal lemma, such as Szemer\'edi's theorem on arithmetic
progressions.
http://arXiv.org/abs/math/0602037
http://front.math.ucdavis.edu/math.CO/0602037
(alternate)
Author(s): Qi Zhang and Huaizhong Zhao
Abstract: The main purpose of this paper is to study the existence of stationary
solution for stochastic partial differential equations. We establish a new
connection between backward doubly stochastic differential equations on
infinite time horizon and the stationary solution of the SPDEs. For this we
study the existence of the solution of the associated BDSDEs on infinite time
horizon and prove it is a stationary viscosity solution of the corresponding
SPDEs.
http://arXiv.org/abs/math/0602054
http://front.math.ucdavis.edu/math.PR/0602054
(alternate)
Author(s): Nathalie Krell (PMA)
Abstract: We consider a mass-conservative fragmentation of the unit interval. The main
purpose of this work is to specify the Hausdorff dimension of the set of
locations having exactly an exponential decay. The study relies on an additive
martingale which arises naturally in this setting, and a class of L\'{e}vy
process constrained to stay in a finite interval.
http://arXiv.org/abs/math/0602065
http://front.math.ucdavis.edu/math.PR/0602065
(alternate)
Author(s): Amine Asselah
Abstract: We obtain large deviations estimates for the self-intersection local times
for a symmetric random walk in dimension 3. Also, we show that the main
contribution to making the self-intersection large, in a time period of length
$n$, comes from sites visited less than some power of $\log(n)$. This is
opposite to the situation in dimensions larger or equal to 5. Finally, we
present two applications of our estimates: (i) to moderate deviations estimates
for the range of a random walk, and (ii) to moderate deviations for random walk
in random sceneries.
http://arXiv.org/abs/math/0602074
http://front.math.ucdavis.edu/math.PR/0602074
(alternate)
Author(s): Andreas Neuenkirch (TU DARMSTADT) and Ivan Nourdin (PMA)
Abstract: In this paper, we derive the exact rate of convergence of some approximation
schemes associated to scalar stochastic differential equations driven by a
fractional Brownian motion with Hurst index H.
http://arXiv.org/abs/math/0601038
http://front.math.ucdavis.edu/math.PR/0601038
(alternate)
Author(s): Alexander Gnedin and Jim Pitman
Abstract: We solve the moment problem for convex distribution functions on $[0,1]$ in
terms of completely alternating sequences. This complements a recent solution
of this problem by Diaconis and Freedman, and relates this work to the
L{\'e}vy-Khintchine formula for the Laplace transform of a subordinator, and to
regenerative composition structures.
http://arXiv.org/abs/math/0602091
http://front.math.ucdavis.edu/math.PR/0602091
(alternate)
Author(s): M. Bramson and T. M. Liggett
Abstract: There has been significant progress recently in our understanding of the
stationary measures of the exclusion process on $Z$. The corresponding
situation in higher dimensions remains largely a mystery. In this paper we give
necessary and sufficient conditions for a product measure to be stationary for
the exclusion process on an arbitrary set, and apply this result to find
examples on $Z^d$ and on homogeneous trees in which product measures are
stationary even when they are neither homogeneous nor reversible. We then begin
the task of narrowing down the possibilities for existence of other stationary
measures for the process on $Z^d$. In particular, we study stationary measures
that are invariant under translations in all directions orthogonal to a fixed
nonzero vector. We then prove a number of convergence results as $t\to\infty$
for the measure of the exclusion process. Under appropriate initial conditions,
we show convergence of such measures to the above stationary measures. We also
employ hydrodynamics to provide further examples of convergence.
http://arXiv.org/abs/math/0602098
http://front.math.ucdavis.edu/math.PR/0602098
(alternate)
Author(s): Julien Berestycki and Nathanael Berestycki and Jason Schweinsberg
Abstract: Coalescents with multiple collisions, also known as $\Lambda$-coalescents,
were introduced by Pitman and Sagitov in 1999. These processes describe the
evolution of particles that undergo stochastic coagulation in such a way that
several blocks can merge at the same time to form a single block. In the case
that the measure $\Lambda$ is the Beta$(2-\alpha,\alpha)$ distribution, they
are also known to describe the genealogies of large populations where a single
individual can produce a large number of offspring. Here we use a recent result
of Birkner et al. to prove that Beta-coalescents can be embedded in continuous
stable random trees, about which much is known due to recent progress of
Duquesne and Le Gall. Our proof is based on a construction of the
Donnelly-Kurtz lookdown process using continuous random trees which is of
independent interest. This produces a number of results concerning the
small-time behavior of Beta-coalescents. Most notably, we recover an almost
sure limit theorem of the authors for the number of blocks at small times, and
give the multifractal spectrum corresponding to the emergence of blocks with
atypical size. Also, we are able to find exact asymptotics for sampling
formulae corresponding to the site frequency spectrum and allele frequency
spectrum associated with mutations in the context of population genetics.
http://arXiv.org/abs/math/0602113
http://front.math.ucdavis.edu/math.PR/0602113
(alternate)
Author(s): Marc Lelarge
Abstract: This paper is devoted to the problem of sample path large deviations for
multidimensional queueing models with feedback. We derive a new version of the
contraction principle where the continuous map is not well-defined on the whole
space: we give conditions under which it allows to identify the rate function.
We illustrate our technique by deriving a large deviation principle for a class
of networks that contains the classical Jackson networks.
http://arXiv.org/abs/math/0602130
http://front.math.ucdavis.edu/math.PR/0602130
(alternate)
Author(s): L. Decreusefond
Abstract: We investigate here the optimal transportation problem on configuration space
for the quadratic cost. It is shown that, as usual, provided that the
corresponding Wasserstein is finite, there exists one unique optimal measure
and that this measure is supported by the graph of the derivative (in the sense
of the Malliavin calculus) of a ``concave'' (in a sense to be defined below)
function. For finite point processes, we give a necessary and sufficient
condition for the Wasserstein distance to be finite.
http://arXiv.org/abs/math/0602134
http://front.math.ucdavis.edu/math.PR/0602134
(alternate)
Author(s): Alice Guionnet (ENS Lyon - UMPA) and \'Edouard Maurel-Segala (ENS Lyon - UMPA)
Abstract: We study several-matrix models and show that when the potential is convex and
a small perturbation of the Gaussian potential, the first order correction to
the free energy can be expressed as a generating function for the enumeration
of maps of genus one. In order to do that, we prove a central limit theorem for
traces of words of the weakly interacting random matrices defined by these
matrix models and show that the variance is a generating function for the
number of planar maps with two vertices with prescribed colored edges.
http://arXiv.org/abs/math/0601040
http://front.math.ucdavis.edu/math.PR/0601040
(alternate)
Author(s): Lauren K. Williams
Abstract: The partially asymmetric exclusion process (PASEP) is an important model from
statistical mechanics which describes a system of interacting particles hopping
left and right on a one-dimensional lattice of n sites. It is partially
asymmetric in the sense that the probability of hopping left is q times the
probability of hopping right. In this paper we prove a close connection between
the PASEP model and the combinatorics of permutation tableaux (certain 0-1
tableaux introduced in a previous paper with Steingrimsson). Namely, we prove
that in the long time limit, the probability that the PASEP model is in a
particular configuration tau is essentially the weight generating function for
permutation tableaux of shape lambda(tau). The proof of this result uses a
result of Derrida et al on the matrix ansatz for the PASEP.
We derive a number of enumerative consequences of the connection between
the PASEP model and permutation tableaux. One consequence is a generating
function for the following (equidistributed) objects: the partition function
for the PASEP model; permutation tableaux of length n+1, enumerated according
to weight; permutations in S_{n+1}, enumerated according to crossings;
permutations in S_{n+1}, enumerated according to occurrences of the generalized
pattern 2-31. Another consequence is a generating function for the subset of
the above objects which is specified by fixing (respectively) a configuration
tau, a shape lambda(tau), a weak excedence set W(tau), or a descent set D(tau).
Note that the equidistribution of permutation tableaux and permutations was
proved in a previous paper of Steingrimsson and the author.
http://arXiv.org/abs/math/0602109
http://front.math.ucdavis.edu/math.CO/0602109
(alternate)
Author(s): Saul Jacka and Abdelkarem Berkaoui and Jon Warren
Abstract: The paper considers trading with proportional transaction costs. We give a
necessary and sufficient condition for $A$, the cone of claims attainable from
zero endowment, to be closed, and show, in general, how to represent its
closure in such a way that it is the cone of claims attainable for zero
endowment, for a different set of trading prices. The new representation obeys
the Fundamental Theorem of Asset Pricing. We then show how to represent claims
and in a final section show how any such setup corresponds to a coherent risk
measure.
http://arXiv.org/abs/math/0602178
http://front.math.ucdavis.edu/math.PR/0602178
(alternate)
Author(s): Erkan Nane
Abstract: We extend generalized isoperimetric-type inequalities to iterated Brownian
motion over several domains in $\RR{R}^{n}$. These kinds of inequalities imply
in particular that for domains of finite volume, the exit distribution and
moments of the first exit time for iterated Brownian motion are maximized with
the ball $D^{*}$ centered at the origin, which has the same volume as $D$
http://arXiv.org/abs/math/0602188
http://front.math.ucdavis.edu/math.PR/0602188
(alternate)
Author(s): M. Grendar
Abstract: In this paper we study the exponential decay of posterior probability of a
set of sources and conditioning by rare sources for both uniform and general
prior distributions of sources. The decay rate is determined by L-divergence
and rare sources from a convex, closed set asymptotically conditionally
concentrate on an L-projection. L-projection on a linear family of sources
belongs to Lambda-family of distributions. The results parallel those of Large
Deviations for Empirical Measures (Sanov's Theorem and Conditional Limit
Theorem).
http://arXiv.org/abs/math/0601048
http://front.math.ucdavis.edu/math.ST/0601048
(alternate)
Author(s): Fr\'{e}d\'{e}ric Paccaut (LAMFA) and Dominique Schneider (LMPA)
Abstract: We study the continuity properties of trajectories for some random series of
functions $\sum a\_kf(\alpha X\_k(\omega))$ where $a\_k$ is a complex sequence,
$X\_k$ a sequence of real independent random variables, $f$ is a real valued
function with period one and summable Fourier coefficients. We obtain almost
sure continuity results for these periodic or almost periodic series for a
large class of functions, where the "almost sure" does not depend on the
function.
http://arXiv.org/abs/math/0602207
http://front.math.ucdavis.edu/math.PR/0602207
(alternate)
Author(s): Andrzej Luczak
Abstract: We show that for a quantum $L^p$-martingale $(X(t))$, $p>2$, there exists a
Doob-Meyer decomposition of the submartingale $(|X(t)|^2)$. A noncommutative
counterpart of a classical process continuous with probability one is
introduced, and a quantum stochastic integral of such a process with respect to
an $L^p$-martingale, $p>2$, is constructed. Using this construction, the
uniqueness of the Doob-Meyer decomposition for a quantum martingale `continuous
with probability one' is proved, and explicit forms of this decomposition and
the quadratic variation process for such a martingale are obtained.
http://arXiv.org/abs/math/0602216
http://front.math.ucdavis.edu/math.OA/0602216
(alternate)
Author(s): G. P. Chistyakov and F. G\"otze
Abstract: Based on a new analytical approach to the definition of additive free
convolution on probability measures on the real line we prove free analogs of
limit theorems for sums for non-identically distributed random variables in
classical Probability Theory.
http://arXiv.org/abs/math/0602219
http://front.math.ucdavis.edu/math.OA/0602219
(alternate)
Author(s): George P. Yanev and Kosto V. Mitov and and Nickolay M. Yanev
Abstract: This paper demonstrates a new regeneration processes technology making use of
positive stable distributions. We study the asymptotic behavior of branching
processes with a randomly controlled migration component. Using the new method,
we confirm some known results and establish new limit theorems that hold in a
more general setting.
http://arXiv.org/abs/math/0602261
http://front.math.ucdavis.edu/math.PR/0602261
(alternate)
Author(s): Soumendra N. Lahiri and A. Chatterjee and and T. Maiti
Abstract: In this paper, we derive a necessary and sufficient condition on the
parameters of the Hypergeometric distribution for weak convergence to a Normal
limit. We establish a Berry-Esseen theorem for the Hypergeometric distribution
solely under this necessary and sufficient condition. We further derive a
nonuniform Berry-Esseen bound where the tails of the difference between the
Hypergeometric and the Normal distribution functions are shown to decay at a
sub-Gaussian rate.
http://arXiv.org/abs/math/0602276
http://front.math.ucdavis.edu/math.PR/0602276
(alternate)
Author(s): Alexander Gnedin
Abstract: We study the best-choice problem for processes which generalise the process
of records from Poisson-paced i.i.d. observations. Under the assumption that
the observer knows distribution of the process and the horizon, we determine
the optimal stopping policy and for a parametric family of problems also derive
an explicit formula for the maximum probability of recognising the last record.
http://arXiv.org/abs/math/0602278
http://front.math.ucdavis.edu/math.PR/0602278
(alternate)
Author(s): A. Faggionato
Abstract: Given a doubly infinite sequence of positive numbers {c_k: k in Z} satisfying
a LLN with limit A, we consider the nearest-neighbor simple exclusion process
on Z where c_k is the probability rate of the jumps between k and k+1. If A is
infinite we require an additional condition corresponding to macroscopic
homogeneity of the medium. By extending a method developed by K. Nagy we show
that the diffusively rescaled process has hydrodynamic behavior described by
the heat equation with diffusion constant 1/A. In particular, the process has
diffusive behavior for finite A and subdiffusive behavior for infinite A.
http://arXiv.org/abs/math/0601076
http://front.math.ucdavis.edu/math.PR/0601076
(alternate)
Author(s): S Satheesh and E Sandhya
Abstract: The family of semi-stable laws is shown to be infinitely divisible and
semi-selfdecomposable. Thus they qualify to model AR(1) schemes. The structure
of AR(1) schemes with semi-stable marginals are explored.
http://arXiv.org/abs/math/0602286
http://front.math.ucdavis.edu/math.PR/0602286
(alternate)
Author(s): Francesco Caravenna and Lo\"ic Chaumont
Abstract: Let {S_n} be a random walk in the domain of attraction of a stable law Y,
i.e. there exists a sequence of positive real numbers (a_n) such that S_n/a_n
converges in law to Y. Our main result is that the rescaled process
(S_[nt]/a_n, t \ge 0), when conditioned to stay positive for all the time,
converges in law (in the functional sense) towards the corresponding stable
Levy process conditioned to stay positive in the same sense. Under some
additional assumptions, we also prove a related invariance principle for the
random walk killed at its first entrance in the negative half-line and
conditioned to die at zero.
http://arXiv.org/abs/math/0602306
http://front.math.ucdavis.edu/math.PR/0602306
(alternate)
Author(s): Antonio Di Crescenzo
Abstract: Parrondo's paradox arises in sequences of games in which a winning
expectation may be obtained by playing the games in a random order, even though
each game in the sequence may be lost when played individually. We present a
suitable version of Parrondo's paradox in reliability theory involving two
systems in series, the units of the first system being less reliable than those
of the second. If the first system is modified so that the distributions of its
new units are mixtures of the previous distributions with equal probabilities,
then under suitable conditions the new system is shown to be more reliable than
the second in the "usual stochastic order" sense.
http://arXiv.org/abs/math/0602308
http://front.math.ucdavis.edu/math.PR/0602308
(alternate)
Author(s): Yan V Fyodorov
Abstract: The paper addresses the calculation of correlation functions of permanental
polynomials of matrices with random entries. By exploiting a convenient contour
integral representation of the matrix permanent some explicit results are
provided for several random matrix ensembles. When compared with the
corresponding formulae for characteristic polynomials, our results show both
striking similarities and interesting differences. Based on these findings, we
conjecture the asymptotic forms of the density of permanental roots in the
complex plane for Gaussian ensembles as well as for the Circular Unitary
Ensemble of large matrix dimension.
http://arXiv.org/abs/math-ph/0602039
http://front.math.ucdavis.edu/math-ph/0602039
(alternate)
Author(s): Joshua Cooper and Benjamin Doerr and Joel Spencer and and Gabor Tardos
Abstract: Jim Propp's P-machine, also known as the "rotor router model" is a simple
deterministic process that simulates a random walk on a graph. Instead of
distributing chips to randomly chosen neighbors, it serves the neighbors in a
fixed order.
We investigate how well this process simulates a random walk. For the graph
being the infinite path, we show that, independent of the starting
configuration, at each time and on each vertex, the number of chips on this
vertex deviates from the expected number of chips in the random walk model by
at most a constant c_1, which is approximately 2.29. For intervals of length L,
this improves to a difference of O(log L), for the L_2 average of a contiguous
set of intervals even to O(sqrt{log L}). All these bounds are tight.
http://arXiv.org/abs/math/0602300
http://front.math.ucdavis.edu/math.CO/0602300
(alternate)
Author(s): Xiaobo Bao and Shanjian Tang
Abstract: This paper introduces the notion of a filtration-consistent dynamic operator
with a floor, by suitably formulating four axioms. It is shown that under some
suitable conditions, a filtration-consistent dynamic operator with a continuous
upper-bounded floor is necessarily represented by the solution of a backward
stochastic differential equation reflected upwards on the floor.
http://arXiv.org/abs/math/0602322
http://front.math.ucdavis.edu/math.PR/0602322
(alternate)
Author(s): Shanjian Tang
Abstract: In this Note, assuming that the generator is uniform Lipschitz in the unknown
variables, we relate the solution of a one dimensional backward stochastic
differential equation with the value process of a stochastic differential game.
Under a domination condition, a filtration-consistent evaluations is also
related to a stochastic differential game. This relation comes out of a min-max
representation for uniform Lipschitz functions as affine functions. The
extension to reflected backward stochastic differential equations is also
included.
http://arXiv.org/abs/math/0602323
http://front.math.ucdavis.edu/math.PR/0602323
(alternate)
Author(s): Celine Jost (University of Helsinki)
Abstract: We proof a connection between the generalized Molchan-Golosov integral
transform and the generalized Mandelbrot-Van Ness integral transform of
fractional Brownian motion (fBm). The former changes fBm of arbitrary Hurst
index K into fBm of index H by integrating over [0,t], whereas the latter
requires integration over (-infty,t].
http://arXiv.org/abs/math/0602356
http://front.math.ucdavis.edu/math.PR/0602356
(alternate)
Author(s): Peter Friz and Nicolas Victoir
Abstract: We consider controlled differential equations and give new estimates for
higher order Euler schemes. Our proofs are inspired by recent work of A. M.
Davie who considers first and second order schemes. In order to implement the
general case we make systematic use of geodesic approximations in the free
nilpotent group. As application, we can control moments of solutions to rough
path differential equations (RDEs) driven by random rough paths with sufficient
integrability and have a criteria for L^q - convergence in the Universal Limit
Theorem. We also obtain Azencott type estimates and asymptotic expansions for
random RDE solution. When specialized to RDEs driven by Enhanced Brownian
motion, we (mildly) improve classic estimates for diffusions in the small time
limit.
http://arXiv.org/abs/math/0602345
http://front.math.ucdavis.edu/math.CA/0602345
(alternate)
Author(s): Serban Teodor Belinschi
Abstract: In this thesis we study convolutions that arise from noncommutative
probability theory. We prove several regularity results for free convolutions,
and for measures in partially defined one-parameter free convolution
semigroups. We discuss connections between Boolean and free convolutions and,
in the last chapter, we prove that any infinitely divisible probability measure
with respect to monotonic additive or multiplicative convolution belongs to a
one-parameter semigroup with respect to the corresponding convolution. Earlier
versions of some of the results in this thesis have already been published,
while some others have been submitted for publication. We have preserved almost
entirely the specific format for PhD theses required by Indiana University.
This adds several unnecessary pages to the document, but we wanted to preserve
the specificity of the document as a PhD thesis at Indiana University.
http://arXiv.org/abs/math/0602343
http://front.math.ucdavis.edu/math.OA/0602343
(alternate)
Author(s): Kalvis M. Jansons and Paul D. Metcalfe
Abstract: We discuss the optimal Markovian coupling before an exponential time of the
Kolmogorov diffusion, and a class of related stochastic control problems in
which the aim is to hit the origin before an exponential time. We provide a
scaling argument for the optimal control in the near field and use rational WKB
approximation to obtain the optimal control in the far field, and compare these
analytical results with numerical experiments.
In some of these optimal control problems, in which the advection velocity
field is bounded, we show that the probability of success field agrees exactly
with its leading-order asymptotic approximation in some areas of the plane, up
to an undetermined multiplicative constant. We conjecture a necessary and
sufficient condition for this behaviour, which is strongly supported by
numerical experiments.
http://arXiv.org/abs/math/0602365
http://front.math.ucdavis.edu/math.PR/0602365
(alternate)
Author(s): Jiagang Ren and Michael R\"ockner and Feng-Yu Wang
Abstract: We present a generalization of Krylov-Rozovskii's result on the existence and
uniqueness of solutions to monotone stochastic differential equations. As an
application, the stochastic generalized porous media and fast diffusion
equations are studied for $\sigma$-finite reference measures, where the drift
term is given by a negative definite operator acting on a time-dependent
function, which belongs to a large class of functions comparable with the
so-called $N$-functions in the theory of Orlicz spaces.
http://arXiv.org/abs/math/0602369
http://front.math.ucdavis.edu/math.PR/0602369
(alternate)
Author(s): Siegried Graf (Universit\"{a}t Passau) and Harald Luschgy and Gilles Pag\`es (PMA)
Abstract: We elucidate the asymptotics of the L^s-quantization error induced by a
sequence of L^r-optimal n-quantizers of a probability distribution P on R^d
when s>r. In particular we show that under natural assumptions, the optimal
rate is preserved as long as s
http://arXiv.org/abs/math/0602381
http://front.math.ucdavis.edu/math.PR/0602381
(alternate)
Author(s): Rosanna Coviello and Francesco Russo
Abstract: In this paper we discuss existence and uniqueness for a one-dimensional time
inhomogeneous stochastic differential equation directed by an $\mathbb
F$-semimartingale $M$ and a finite cubic variation process $\xi$ which has the
structure $Q + R$ where $Q$ is a finite quadratic variation process and $R$ is
{\it{strongly predictable}} in some technical sense: that condition implies in
particular that $R$ is \textit{weak Dirichlet}, and it is fulfilled, for
instance, when $R$ is independent of $M$. The method is based on a
transformation which reduces the {{\it diffusion}} coefficient multiplying
$\xi$ to 1. We use generalized It\^o and It\^o-Wentzell type formulae. A
similar method allows to discuss existence and uniqueness theorem when $\xi$ is
a H\"older continuous process and $\sigma$ is only H\"older in space. Using an
It\^o formula for {\it{reversible}} semimartingales we also show existence of a
solution when $\xi$ is a Brownian motion and $\sigma$ is only continuous.
http://arXiv.org/abs/math/0602384
http://front.math.ucdavis.edu/math.PR/0602384
(alternate)
Author(s): Robert O. Bauer
Abstract: We study the probability that chordal $\text{SLE}_{8/3}$ in the unit disk
from $\exp(ix)$ to 1 avoids the disk of radius $q$ centered at zero. We find
the initial/boundary-value problem satisfied by this probability as a function
of $x$ and $a=\ln q$, and show that asymptotically as $q$ tends to one this
probability decays like $\exp(-c/(1-q))$ with $c=5\pi^2/16$. We also give a
representation of this probability as a functional of a Legendre process.
http://arXiv.org/abs/math/0602391
http://front.math.ucdavis.edu/math.PR/0602391
(alternate)
Author(s): David R. Bickel
Abstract: Prior information can be incorporated into a p-value or CI by combining the
confidence distribution (CD) from the observed data with one or more
independent CDs representing well-calibrated expert opinion. The first CD may
be an objective Bayes posterior distribution.
http://arXiv.org/abs/math/0602377
http://front.math.ucdavis.edu/math.ST/0602377
(alternate)
Author(s): Markus Fischer and Markus Reiss
Abstract: As a main step in the numerical solution of control problems in continuous
time, the controlled process is approximated by sequences of controlled Markov
chains, thus discretising time and space. A new feature in this context is to
allow for delay in the dynamics. The existence of an optimal strategy with
respect to the cost functional can be guaranteed in the class of relaxed
controls. Weak convergence of the approximating extended Markov chains to the
original process together with convergence of the associated optimal strategies
is established.
http://arXiv.org/abs/math/0602385
http://front.math.ucdavis.edu/math.OC/0602385
(alternate)
Author(s): Valentin Konakov
Abstract: We consider triangular arrays of Markov chains that converge weakly to a
diffusion process. Local limit theorems for transition densities are proved.
The observation time [0,T] may be fixed or lim n T = 0, where nh = T and h is a
mesh between two neighboring observation points.
http://arXiv.org/abs/math/0602429
http://front.math.ucdavis.edu/math.PR/0602429
(alternate)
Author(s): Valentin Konakov
Abstract: We consider triangular arrays of Markov chains that converge weakly to a
diffusio process. Edgeworth type expansions of third order for transition
densities are proved. This is done for time horizons that converge to 0. For
this purpose we represent the transition density as a functional of densities
of sums of i.i.d. variables. This will be done by application of the parametrix
method. Then we apply Edgeworth expansions to the densities. The resulting
series gives our Edgeworth-type expansion for the transition density of Markov
chains. The research is motivated by applications to high frequency data that
are available on a very fine grid but are approximated by a diffusion model on
a more rough grid.
http://arXiv.org/abs/math/0602430
http://front.math.ucdavis.edu/math.PR/0602430
(alternate)
Author(s): Masaki Izumi and Sergey Neshveyev and Rui Okayasu
Abstract: We consider the harmonic measure on the Gromov boundary of a nonamenable
hyperbolic group defined by a finite range random walk on the group, and study
the corresponding orbit equivalence relation on the boundary. It is known to be
always amenable and of type III. We determine its ratio set by showing that it
is generated by certain values of the Martin kernel. In particular, we show
that the equivalence relation is never of type III_0.
http://arXiv.org/abs/math/0602409
http://front.math.ucdavis.edu/math.DS/0602409
(alternate)
Author(s): Bruno Bouchard and Huy\^en Pham
Abstract: We consider a general discrete-time financial market with proportional
transaction costs as in [Kabanov, Stricker and R\'{a}sonyi Finance and
Stochastics 7 (2003) 403--411] and [Schachermayer Math. Finance 14 (2004)
19--48]. In addition to the usual investment in financial assets, we assume
that the agents can invest part of their wealth in industrial projects that
yield a nonlinear random return. We study the problem of maximizing the utility
of consumption on a finite time period. The main difficulty comes from the
nonlinearity of the nonfinancial assets' return. Our main result is to show
that existence holds in the utility maximization problem. As an intermediary
step, we prove the closedness of the set $A_T$ of attainable claims under a
robust no-arbitrage property similar to the one introduced in [Schachermayer
Math. Finance 14 (2004) 19--48] and further discussed in [Kabanov, Stricker and
R\'{a}sonyi Finance and Stochastics 7 (2003) 403--411]. This allows us to
provide a dual formulation for $A_T$.
http://arXiv.org/abs/math/0602451
http://front.math.ucdavis.edu/math.PR/0602451
(alternate)
Author(s): Patrick Cheridito and H. Mete Soner and Nizar Touzi
Abstract: We study the small time path behavior of double stochastic integrals of the
form $\int_0^t(\int_0^rb(u) dW(u))^T dW(r)$, where $W$ is a $d$-dimensional
Brownian motion and $b$ is an integrable progressively measurable stochastic
process taking values in the set of $d\times d$-matrices. We prove a law of the
iterated logarithm that holds for all bounded progressively measurable $b$ and
give additional results under continuity assumptions on $b$. As an application,
we discuss a stochastic control problem that arises in the study of the
super-replication of a contingent claim under gamma constraints.
http://arXiv.org/abs/math/0602453
http://front.math.ucdavis.edu/math.PR/0602453
(alternate)
Author(s): Henrik Hult and Filip Lindskog and Thomas Mikosch and Gennady Samorodnitsky
Abstract: We extend classical results by A. V. Nagaev [Izv. Akad. Nauk UzSSR Ser.
Fiz.--Mat. Nauk 6 (1969) 17--22, Theory Probab. Appl. 14 (1969) 51--64,
193--208] on large deviations for sums of i.i.d. regularly varying random
variables to partial sum processes of i.i.d. regularly varying vectors. The
results are stated in terms of a heavy-tailed large deviation principle on the
space of c\`{a}dl\`{a}g functions. We illustrate how these results can be
applied to functionals of the partial sum process, including ruin probabilities
for multivariate random walks and long strange segments. These results make
precise the idea of heavy-tailed large deviation heuristics: in an asymptotic
sense, only the largest step contributes to the extremal behavior of a
multivariate random walk.
http://arXiv.org/abs/math/0602460
http://front.math.ucdavis.edu/math.PR/0602460
(alternate)
Author(s): Bruno Bouchard and Nicole El Karoui and Nizar Touzi
Abstract: We study a maturity randomization technique for approximating optimal control
problems. The algorithm is based on a sequence of control problems with random
terminal horizon which converges to the original one. This is a generalization
of the so-called Canadization procedure suggested by Carr [Review of Financial
Studies II (1998) 597--626] for the fast computation of American put option
prices. In addition to the original application of this technique to optimal
stopping problems, we provide an application to another problem in finance,
namely the super-replication problem under stochastic volatility, and we show
that the approximating value functions can be computed explicitly.
http://arXiv.org/abs/math/0602462
http://front.math.ucdavis.edu/math.PR/0602462
(alternate)
Author(s): Liqing Yan
Abstract: A Milstein-type scheme was proposed to improve the rate of convergence of its
approximation of the solution to a stochastic differential equation driven by a
vector of continuous semimartingales. A necessary and sufficient condition was
provided for this rate to be $1/n$ when the SDE is driven by a vector of
continuous local martingales, or continuous semimartingales under an additional
assumption on their finite variation part. The asymptotic behavior (weak
convergence) of the normalized error processes was also studied.
http://arXiv.org/abs/math/0602465
http://front.math.ucdavis.edu/math.PR/0602465
(alternate)
Author(s): A.C.D. van Enter and C. Kuelske
Abstract: In this contribution we discuss the role disordered (or random) systems have
played in the study of non-Gibbsian measures. This role has two main aspects,
the distinction between which has not always been fully clear:
1) {\em From} disordered systems: Disordered systems can be used as a tool;
analogies with, as well as results and methods from the study of random systems
can be employed to investigate non-Gibbsian properties of a variety of measures
of physical and mathematical interest.
2) {\em Of} disordered systems: Non-Gibbsianness is a property of various
(joint) measures describing quenched disordered systems.
We discuss and review this distinction and a number of results related to
these issues. Moreover, we discuss the mean-field version of the non-Gibbsian
property, and present some ideas how a Kac limit approach might connect the
finite-range and the mean-field non-Gibbsian properties.
http://arXiv.org/abs/math-ph/0602047
http://front.math.ucdavis.edu/math-ph/0602047
(alternate)
Author(s): D. T. Hristopulos
Abstract: The spatial structure of fluctuations in spatially inhomogeneous processes
can be modeled in terms of Gibbs random fields. A local low energy estimator
(LLEE) is proposed for the interpolation (prediction) of such processes at
points where observations are not available. The LLEE approximates the spatial
dependence of the data and the unknown values at the estimation points by
low-lying excitations of a suitable energy functional. It is shown that the
LLEE is a linear, unbiased, non-exact estimator. In addition, an expression for
the uncertainty (standard deviation) of the estimate is derived.
http://arXiv.org/abs/physics/0510035
http://front.math.ucdavis.edu/physics/0510035
(alternate)
Author(s): Steven N. Evans
Abstract: We study the simultaneous zeros of a random family of $d$ polynomials in $d$
variables over the $p$-adic numbers. For a family of natural models, we obtain
an explicit constant for the expected number of zeros that lie in the $d$-fold
Cartesian product of the $p$-adic integers. This expected value, which is \[ (1
+ p^{-1} + p^{-2} + ... + p^{-d})^{-1} \] for the simplest model, is
independent of the degree of the polynomials.
http://arXiv.org/abs/math/0602478
http://front.math.ucdavis.edu/math.PR/0602478
(alternate)
Author(s): Martin Hairer and Jonathan C. Mattingly
Abstract: We develop a general method that allows to show the existence of spectral
gaps for Markov semigroups on Banach spaces. Unlike most previous work, the
type of norm we consider for this analysis is neither a weighted supremum norm
nor an L^p-type norm, but involves the derivative of the observable as well and
hence can be seen as a type of 1--Wasserstein distance. This turns out to be a
suitable approach for infinite-dimensional spaces where the usual Harris or
Doeblin conditions, which are geared to total variation convergence, regularly
fail to hold. In the first part of this paper, we consider semigroups that have
uniform behaviour which one can view as an extension of Doeblin's condition. We
then proceed to study situations where the behaviour is not so uniform, but the
system has a suitable Lyapunov structure, leading to a type of Harris
condition. We finally show that the latter condition is satisfied by the
two-dimensional stochastic Navier-Stokers equations, even in situations where
the forcing is extremely degenerate. Using the convergence result, we show shat
the stochastic Navier-Stokes equations' invariant measures depend continuously
on the viscosity and the structure of the forcing.
http://arXiv.org/abs/math/0602479
http://front.math.ucdavis.edu/math.PR/0602479
(alternate)
Author(s): Michael A. Kouritzin and Wei Sun
Abstract: Herein, we analyze an efficient branching particle method for asymptotic
solutions to a class of continuous-discrete filtering problems. Suppose that
$t\to X_t$ is a Markov process and we wish to calculate the measure-valued
process $t\to\mu_t(\cdot)\doteq P\{X_t\in \cdot|\sigma\{Y_{t_k}, t_k\leq
t\}\}$, where $t_k=k\epsilon$ and $Y_{t_k}$ is a distorted, corrupted, partial
observation of $X_{t_k}$. Then, one constructs a particle system with
observation-dependent branching and $n$ initial particles whose empirical
measure at time $t$, $\mu_t^n$, closely approximates $\mu_t$. Each particle
evolves independently of the other particles according to the law of the signal
between observation times $t_k$, and branches with small probability at an
observation time. For filtering problems where $\epsilon$ is very small, using
the algorithm considered in this paper requires far fewer computations than
other algorithms that branch or interact all particles regardless of the value
of $\epsilon$. We analyze the algorithm on L\'{e}vy-stable signals and give
rates of convergence for $E^{1/2}\{\|\mu^n_t-\mu_t\|_{\gamma}^2\}$, where
$\Vert\cdot\Vert_{\gamma}$ is a Sobolev norm, as well as related convergence
results.
http://arXiv.org/abs/math/0602488
http://front.math.ucdavis.edu/math.PR/0602488
(alternate)
Author(s): J. M. Harrison and R. J. Williams
Abstract: We consider a dynamic control problem associated with a generalized Brownian
network, the objective being to minimize expected discounted cost over an
infinite planning horizon. In this Brownian control problem (BCP), both the
system manager's control and the associated cumulative cost process may be
locally of unbounded variation. Due to this aspect of the cost process, both
the precise statement of the problem and its analysis involve delicate
technical issues. We show that the BCP is equivalent, in a certain sense, to a
reduced Brownian control problem (RBCP) of lower dimension. The RBCP is a
singular stochastic control problem, in which both the controls and the
cumulative cost process are locally of bounded variation.
http://arXiv.org/abs/math/0602495
http://front.math.ucdavis.edu/math.PR/0602495
(alternate)
Author(s): Michel Benaim and Raphael Rossignol
Abstract: We prove a new functional inequality for a countable product of Gaussian
measures which is the exact counterpart of an inequality by Talagrand for
products of Bernoulli measures. This inequality improves on the classical
Poincare inequality for Gaussian measures. As an application, we prove that
First Passage Percolation has sublinear variance when the edge times
distribution belongs to a wide class of continuous distributions, including the
exponential one. This extends a result by Benjamini, Kalai and Schramm, valid
for positive Bernoulli edge times.
http://arXiv.org/abs/math/0602496
http://front.math.ucdavis.edu/math.PR/0602496
(alternate)
Author(s): Fran\c{c}oise P\`{e}ne
Abstract: We show how Rio's method [Probab. Theory Related Fields 104 (1996) 255--282]
can be adapted to establish a rate of convergence in ${\frac{1}{\sqrt{n}}}$ in
the multidimensional central limit theorem for some stationary processes in the
sense of the Kantorovich metric. We give two applications of this general
result: in the case of the Knudsen gas and in the case of the Sinai billiard.
http://arXiv.org/abs/math/0602501
http://front.math.ucdavis.edu/math.PR/0602501
(alternate)
Author(s): Emmanuel Gobet (LMC - IMAG) and C\'{e}line Labart (CMAP)
Abstract: We study the error induced by the time discretization of a decoupled
forward-backward stochastic differential equations $(X,Y,Z)$. The forward
component $X$ is the solution of a Brownian stochastic differential equation
and is approximated by a Euler scheme $X^N$ with $N$ time steps. The backward
component is approximated by a backward scheme. Firstly, we prove that the
errors $(Y^N-Y,Z^N-Z)$ measured in the strong $L\_p$-sense ($p \geq 1$) are of
order $N^{-1/2}$ (this generalizes the results by Zhang 2004). Secondly, an
error expansion is derived: surprisingly, the first term is proportional to
$X^N-X$ while residual terms are of order $N^{-1}$.
http://arXiv.org/abs/math/0602503
http://front.math.ucdavis.edu/math.PR/0602503
(alternate)
Author(s): Jan Poland and Marcus Hutter
Abstract: The Minimum Description Length principle for online sequence
estimation/prediction in a proper learning setup is studied. If the underlying
model class is discrete, then the total expected square loss is a particularly
interesting performance measure: (a) this quantity is finitely bounded,
implying convergence with probability one, and (b) it additionally specifies
the convergence speed. For MDL, in general one can only have loss bounds which
are finite but exponentially larger than those for Bayes mixtures. We show that
this is even the case if the model class contains only Bernoulli distributions.
We derive a new upper bound on the prediction error for countable Bernoulli
classes. This implies a small bound (comparable to the one for Bayes mixtures)
for certain important model classes. We discuss the application to Machine
Learning tasks such as classification and hypothesis testing, and
generalization to countable classes of i.i.d. models.
http://arXiv.org/abs/math/0602505
http://front.math.ucdavis.edu/math.ST/0602505
(alternate)
Author(s): Jason Schweinsberg
Abstract: Let $x$ and $y$ be points chosen uniformly at random from $\Z_n^4$, the
four-dimensional discrete torus with side length $n$. We show that the length
of the loop-erased random walk from $x$ to $y$ is of order $n^2 (\log
n)^{1/6}$, resolving a conjecture of Benjamini and Kozma. We also show that the
scaling limit of the uniform spanning tree on $\Z_n^4$ is the Brownian
continuum random tree of Aldous. Our proofs use the techniques developed by
Peres and Revelle, who studied the scaling limits of the uniform spanning tree
on a large class of finite graphs that includes the $d$-dimensional discrete
torus for $d \geq 5$, in combination with results of Lawler concerning
intersections of four-dimensional random walks.
http://arXiv.org/abs/math/0602515
http://front.math.ucdavis.edu/math.PR/0602515
(alternate)
Author(s): Adrian D. Banner and Robert Fernholz and Ioannis Karatzas
Abstract: Atlas-type models are constant-parameter models of uncorrelated stocks for
equity markets with a stable capital distribution, in which the growth rates
and variances depend on rank. The simplest such model assigns the same,
constant variance to all stocks; zero rate of growth to all stocks but the
smallest; and positive growth rate to the smallest, the Atlas stock. In this
paper we study the basic properties of this class of models, as well as the
behavior of various portfolios in their midst. Of particular interest are
portfolios that do not contain the Atlas stock.
http://arXiv.org/abs/math/0602521
http://front.math.ucdavis.edu/math.PR/0602521
(alternate)
Author(s): Alexandros Beskos and Gareth O. Roberts
Abstract: We describe a new, surprisingly simple algorithm, that simulates exact sample
paths of a class of stochastic differential equations. It involves rejection
sampling and, when applicable, returns the location of the path at a random
collection of time instances. The path can then be completed without further
reference to the dynamics of the target process.
http://arXiv.org/abs/math/0602523
http://front.math.ucdavis.edu/math.PR/0602523
(alternate)
Author(s): Pierre Del Moral and Josselin Garnier
Abstract: In this paper an original interacting particle system approach is developed
for studying Markov chains in rare event regimes. The proposed particle system
is theoretically studied through a genealogical tree interpretation of
Feynman--Kac path measures. The algorithmic implementation of the particle
system is presented. An estimator for the probability of occurrence of a rare
event is proposed and its variance is computed, which allows to compare and to
optimize different versions of the algorithm. Applications and numerical
implementations are discussed. First, we apply the particle system technique to
a toy model (a Gaussian random walk), which permits to illustrate the
theoretical predictions. Second, we address a physically relevant problem
consisting in the estimation of the outage probability due to polarization-mode
dispersion in optical fibers.
http://arXiv.org/abs/math/0602525
http://front.math.ucdavis.edu/math.PR/0602525
(alternate)
Author(s): Rami Atar
Abstract: A multiclass queueing system is considered, with heterogeneous service
stations, each consisting of many servers with identical capabilities. An
optimal control problem is formulated, where the control corresponds to
scheduling and routing, and the cost is a cumulative discounted functional of
the system's state. We examine two versions of the problem: ``nonpreemptive,''
where service is uninterruptible, and ``preemptive,'' where service to a
customer can be interrupted and then resumed, possibly at a different station.
We study the problem in the asymptotic heavy traffic regime proposed by Halfin
and Whitt, in which the arrival rates and the number of servers at each station
grow without bound. The two versions of the problem are not, in general,
asymptotically equivalent in this regime, with the preemptive version showing
an asymptotic behavior that is, in a sense, much simpler. Under appropriate
assumptions on the structure of the system we show: (i) The value function for
the preemptive problem converges to $V$, the value of a related diffusion
control problem. (ii) The two versions of the problem are asymptotically
equivalent, and in particular nonpreemptive policies can be constructed that
asymptotically achieve the value $V$. The construction of these policies is
based on a Hamilton--Jacobi--Bellman equation associated with $V$.
http://arXiv.org/abs/math/0602526
http://front.math.ucdavis.edu/math.PR/0602526
(alternate)
Author(s): M. Hairer and A. M. Stuart and and J. Voss
Abstract: In many applications it is important to be able to sample paths of SDEs
conditional on observations of various kinds. This paper studies SPDEs which
solve such sampling problems. The SPDE may be viewed as an infinite dimensional
analogue of the Langevin SDE used in finite dimensional sampling. In this paper
nonlinear SDEs, leading to nonlinear SPDEs for the sampling, are studied. In
addition, a class of preconditioned SPDEs is studied, found by applying a
Green's operator to the SPDE in such a way that the invariant measure remains
unchanged; such infinite dimensional evolution equations are important for the
development of practical algorithms for sampling infinite dimensional problems.
The resulting SPDEs provide several significant challenges in the theory of
SPDEs. The two primary ones are the presence of nonlinear boundary conditions,
involving first order derivatives, and a loss of the smoothing property in the
case of the pre-conditioned SPDEs. These challenges are overcome and a theory
of existence, uniqueness and ergodicity developed in sufficient generality to
subsume the sampling problems of interest to us. The Gaussian theory developed
in Part~I of this paper considers Gaussian SDEs, leading to linear Gaussian
SPDEs for sampling. This Gaussian theory is used as the basis for deriving
nonlinear SPDEs which effect the desired sampling in the nonlinear case, via a
change of measure.
http://arXiv.org/abs/math/0601092
http://front.math.ucdavis.edu/math.PR/0601092
(alternate)
Author(s): Ahmed Kebaier
Abstract: We study the approximation of $\mathbb{E}f(X_T)$ by a Monte Carlo algorithm,
where $X$ is the solution of a stochastic differential equation and $f$ is a
given function. We introduce a new variance reduction method, which can be
viewed as a statistical analogue of Romberg extrapolation method. Namely, we
use two Euler schemes with steps $\delta$ and $\delta^{\beta},0<\beta<1$. This
leads to an algorithm which, for a given level of the statistical error, has a
complexity significantly lower than the complexity of the standard Monte Carlo
method. We analyze the asymptotic error of this algorithm in the context of
general (possibly degenerate) diffusions. In order to find the optimal $\beta$
(which turns out to be $\beta=1/2$), we establish a central limit type theorem,
based on a result of Jacod and Protter for the asymptotic distribution of the
error in the Euler scheme. We test our method on various examples. In
particular, we adapt it to Asian options. In this setting, we have a CLT and,
as a by-product, an explicit expansion of the discretization error.
http://arXiv.org/abs/math/0602529
http://front.math.ucdavis.edu/math.PR/0602529
(alternate)
Author(s): M. De Donno and M. Pratelli
Abstract: We introduce a theory of stochastic integration with respect to a family of
semimartingales depending on a continuous parameter, as a mathematical
background to the theory of bond markets. We apply our results to the problem
of super-replication and utility maximization from terminal wealth in a bond
market. Finally, we compare our approach to those already existing in
literature.
http://arXiv.org/abs/math/0602532
http://front.math.ucdavis.edu/math.PR/0602532
(alternate)
Author(s): Guy Cohen and Christophe Cuny
Abstract: We study random exponential sums of the form $\sum_{k=1}^nX_k\times\ex
p\{i(\lambda_k^{(1)}t_1+...+\lambda_k^{(s)}t_s)\}$, where $\{X_n\}$ is a
sequence of random variables and $\{\lambda_n^{(i)}:1\leq i\leq s\}$ are
sequences of real numbers. We obtain uniform estimates (on compact sets) of
such sums, for independent centered $\{X_n\}$ or bounded $\{X_n\}$ satisfying
some mixing conditions. These results generalize recent results of Weber [Math.
Inequal. Appl. 3 (2000) 443--457] and Fan and Schneider [Ann. Inst. H.
Poincar\'{e} Probab. Statist. 39 (2003) 193--216] in several directions. As
applications we derive conditions for uniform convergence of these sums on
compact sets. We also obtain random ergodic theorems for finitely many
commuting measure-preserving point transformations of a probability space.
Finally, we show how some of our results allow to derive the Wiener--Wintner
property (introduced by Assani [Ergodic Theory Dynam. Systems 23 (2003)
1637--1654]) for certain functions on certain dynamical systems.
http://arXiv.org/abs/math/0602543
http://front.math.ucdavis.edu/math.PR/0602543
(alternate)
Author(s): Jonathan E. Taylor
Abstract: In this paper we consider probabilistic analogues of some classical integral
geometric formulae: Weyl--Steiner tube formulae and the Chern--Federer
kinematic fundamental formula. The probabilistic building blocks are smooth,
real-valued random fields built up from i.i.d. copies of centered,
unit-variance smooth Gaussian fields on a manifold $M$. Specifically, we
consider random fields of the form $f_p=F(y_1(p),...,y_k(p))$ for $F\in
C^2(\mathbb{R}^k;\mathbb{R})$ and $(y_1,...,y_k)$ a vector of $C^2$ i.i.d.
centered, unit-variance Gaussian fields. The analogue of the Weyl--Steiner
formula for such Gaussian-related fields involves a power series expansion for
the Gaussian, rather than Lebesgue, volume of tubes: that is, power series
expansions related to the marginal distribution of the field $f$. The formal
expansions of the Gaussian volume of a tube are of independent geometric
interest. As in the classical Weyl--Steiner formulae, the coefficients in these
expansions show up in a kinematic formula for the expected Euler
characteristic, $\chi$, of the excursion sets $M\cap f^{-1}[u,+\infty)=M\cap
y^{-1}(F^{-1}[u,+\infty))$ of the field $f$. The motivation for studying the
expected Euler characteristic comes from the well-known approximation
$\mathbb{P}[\sup_{p\in M}f(p)\geq u]\simeq\mathbb{E}[\chi(f^{-1}[u,+\infty))]$.
http://arXiv.org/abs/math/0602545
http://front.math.ucdavis.edu/math.PR/0602545
(alternate)
Author(s): Fabrice Baudoin and David Nualart
Abstract: We study the two-dimensional fractional Brownian motion with Hurst parameter
$H>{1/2}$. In particular, we show, using stochastic calculus, that this process
admits a skew-product decomposition and deduce from this representation some
asymptotic properties of the motion.
http://arXiv.org/abs/math/0602547
http://front.math.ucdavis.edu/math.PR/0602547
(alternate)
Author(s): Jean-Fran\c{c}ois Collet (JAD) and Florent Malrieu (IRMAR)
Abstract: We investigate the dissipativity properties of a class of scalar second order
parabolic partial differential equations with time-dependent coefficients. We
provide explicit condition on the drift term which ensure that the relative
entropy of one particular orbit with respect to some other one decreases to
zero. The decay rate is obtained explicitly by the use of a Sobolev logarithmic
inequality for the associated semigroup, which is derived by an adaptation of
Bakry's $\Gamma-$ calculus. As a byproduct, the systematic method for
constructing entropies which we propose here also yields the well-known
intermediate asymptotics for the heat equation in a very quick way, and without
having to rescale the original equation.
http://arXiv.org/abs/math/0602548
http://front.math.ucdavis.edu/math.PR/0602548
(alternate)
Author(s): Iosif Pinelis
Abstract: Let $\BS_1,...,\BS_n$ be independent identically distributed random variables
each having the standardized Bernoulli distribution with parameter $p\in(0,1)$.
Let $m_*(p):=(1+p+2p^2)/(2\sqrt{p-p^2}+4p^2)$ if $0
http://arXiv.org/abs/math/0602556
http://front.math.ucdavis.edu/math.PR/0602556
(alternate)
Author(s): L. Bertini and A. De Sole and D. Gabrielli and G. Jona-Lasinio and C. Landim
Abstract: We present a review of recent work on the statistical mechanics of non
equilibrium processes based on the analysis of large deviations properties of
microscopic systems. Stochastic lattice gases are non trivial models of such
phenomena and can be studied rigorously providing a source of challenging
mathematical problems. In this way, some principles of wide validity have been
obtained leading to interesting physical consequences.
http://arXiv.org/abs/math/0602557
http://front.math.ucdavis.edu/math.PR/0602557
(alternate)
Author(s): Mark Rudelson and Roman Vershynin
Abstract: We want to exactly reconstruct a sparse signal f (a vector in R^n of small
support) from few linear measurements of f (inner products with some fixed
vectors). A nice and intuitive reconstruction by Linear Programming has been
advocated since 80-ies by Dave Donoho and his collaborators. Namely, one can
relax the reconstruction problem, which is highly nonconvex, to a convex
problem -- and, moreover, to a linear program. However, when is exactly the
reconstruction problem equivalent to its convex relaxation is an open question.
Recent work of many authors shows that the number of measurements k(r,n) needed
to exactly reconstruct any r-sparse signal f of length n (a vector in R^n of
support r) from its linear measurements with the convex relaxation method is
usually O(r polylog(n)). However, known estimates of the number of measurements
k(r,n) involve huge constants, in spite of very good performance of the
algorithms in practice. In this paper, we consider random Gaussian measurements
and random Fourier measurements (a frequency sample of f). For Gaussian
measurements, we prove the first guarantees with reasonable constants: k(r,n) <
12 r (2 + log(n/r)), which is optimal up to constants. For Fourier
measurements, we prove the best known bound k(r,n) = O(r log(n) . log^2(r)
log(r log n)), which is optimal within the log log n and log^2 r factors. Our
arguments are based on the technique of Geometric Functional Analysis and
Probability in Banach spaces, in particular of Mark Rudelson's sampling method
for random vectors in the isotropic position.
http://arXiv.org/abs/math/0602559
http://front.math.ucdavis.edu/math.NA/0602559
(alternate)
Author(s): M. Hairer and A. M. Stuart and J. Voss and and P. Wiberg
Abstract: In many applications it is important to be able to sample paths of SDEs
conditional on observations of various kinds. This paper studies SPDEs which
solve such sampling problems. The SPDE may be viewed as an infinite dimensional
analogue of the Langevin SDE used in finite dimensional sampling. Here the
theory is developed for conditioned Gaussian processes for which the resulting
SPDE is linear. Applications include the Kalman-Bucy filter/smoother. A
companion paper studies the nonlinear case, building on the linear analysis
provided here.
http://arXiv.org/abs/math/0601095
http://front.math.ucdavis.edu/math.PR/0601095
(alternate)
Author(s): Nadine Guillotin-Plantard and Arnaud Le Ny
Abstract: We study the asymptotic behavior of the simple random walk on oriented
versions of $\mathbb{Z}^2$. The considered lattices are not directed on the
vertical axis but unidirectional on the horizontal one, with random
orientations whose distributions are generated by a dynamical system. We find a
sufficient condition on the smoothness of the generation for the transience of
the simple random walk on almost every such oriented lattices, and as an
illustration we provide a wide class of examples of inhomogeneous or correlated
distributions of the orientations. For ergodic dynamical systems, we also prove
a strong law of large numbers and, in the particular case of i.i.d.
orientations, we solve an open problem and prove a functional limit theorem in
a corresponding space D of cadlag functions, with an unconventional
normalization.
http://arXiv.org/abs/math/0601102
http://front.math.ucdavis.edu/math.PR/0601102
(alternate)
Author(s): Heng-Qing Ye and David D. Yao
Abstract: We study a stochastic network that consists of a set of servers processing
multiple classes of jobs. Each class of jobs requires a concurrent occupancy of
several servers while being processed, and each server is shared among the job
classes in a head-of-the-line processor-sharing mechanism. The allocation of
the service capacities is a real-time control mechanism: in each network state,
the control is the solution to an optimization problem that maximizes a general
utility function. Whereas this resource control optimizes in a ``greedy''
fashion, with respect to each state, we establish its asymptotic optimality in
terms of (a) deriving the fluid and diffusion limits of the network under this
control, and (b) identifying a cost function that is minimized in the diffusion
limit, along with a characterization of the so-called fixed point state of the
network.
http://arXiv.org/abs/math/0601088
http://front.math.ucdavis.edu/math.OC/0601088
(alternate)
Author(s): Rapha\"el Rossignol
Abstract: We investigate the threshold widths of some symmetric properties which range
asymptotically between 1/\sqrt{n} and 1/(log n). These properties are built
using a combination of failure sets arising from reliability theory. This
combination of sets is simply called a product. Some general results on the
threshold width of the product of two sets A and B in terms of the threshold
locations and widths of A and B are provided.
http://arXiv.org/abs/math/0601116
http://front.math.ucdavis.edu/math.PR/0601116
(alternate)
Author(s): Charles Bordenave
Abstract: On a locally finite point set, a navigation defines a path through the point
set from a point to an other. The set of paths leading to a given point defines
a tree, the navigation tree. In this article, we analyze the properties of the
navigation tree when the point set is a Poisson point process on $\R^d$. We
examine the distribution of stable functionals, the local weak convergence of
the navigation tree, the asymptotic average of a functional along a path, the
shape of the navigation tree and its topological ends. We illustrate our work
in the small world graphs, and new results are established. This work is
motivated by applications in computational geometry and in self-organizing
networks.
http://arXiv.org/abs/math/0601122
http://front.math.ucdavis.edu/math.PR/0601122
(alternate)
Author(s): Jeremy Quastel
Abstract: We consider a system of random walks in a random environment interacting via
exclusion. The model is reversible with respect to a family of disordered
Bernoulli measures. Assuming some weak mixing conditions, it is shown that
under diffusive scaling the system has a deterministic hydrodynamic limit which
holds for almost every realization of the environment. The limit is a nonlinear
diffusion equation with diffusion coefficient given by a variational formula.
The model is nongradient and the method used is the ``long jump'' variation of
the standard nongradient method, which is a type of renormalization. The proof
is valid in all dimensions.
http://arXiv.org/abs/math/0601124
http://front.math.ucdavis.edu/math.PR/0601124
(alternate)
Author(s): Roman Vershynin
Abstract: This note deals with a problem of the probabilistic Ramsey theory. Given a
linear operator T on a Hilbert space with an orthogonal basis, we define the
isomorphic structure Sigma(T) as the family of all finite subsets of the basis
such that T restricted to their span is a nice isomorphism. We give an optimal
bound on the size of Sigma(T). This improves and extends in several ways the
principle of restricted invertibility due to Bourgain and Tzafriri. With an
appropriate notion of randomness, we obtain a randomized principle of
restricted invertibility.
http://arXiv.org/abs/math/0601112
http://front.math.ucdavis.edu/math.FA/0601112
(alternate)
Author(s): Lev Sakhnovich
Abstract: We investigate the asymptotic behavior of sample functions of stable
processes when $t | |
