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Probability Abstracts 89
This document contains abstracts 3647-3795.
They have been mailed on November 7, 2005.
Author(s): Francoise Pene
Abstract: The motivation of this work is the study of the error term
e_t^{\epsilon}(x,\omega) in the averaging method for differential equations
perturbed by a dynamical system. Results of convergence in distribution for
(\frac{e_t^{\epsilon}(x,\cdot)}{\sqrt\epsilon})_{\epsilon>0} have been
established in Khas'minskii [Theory Probab. Appl. 11 (1966) 211-228], Kifer
[Ergodic Theory Dynamical Systems 15 (1995) 1143-1172] and P\`ene [ESAIM
Probab. Statist. 6 (2002) 33-88]. We are interested here in the question of the
rate of convergence in distribution of the family of random variables
(\frac{e_t^{\epsilon}(x,\cdot)}{\sqrt\epsilon})_{\epsilon>0} when \epsilon goes
to 0 (t>0 and x\inR^d being fixed). We will make an assumption of multiple
decorrelation property (satisfied in several situations). We start by
establishing a simpler result: the rate of convergence in the central limit
theorem for regular multidimensional functions. In this context, we prove a
result of convergence in distribution with rate of convergence in
O(n^{-1/2+\alpha}) for all \alpha>0 (for the Prokhorov metric). This result can
be seen as an extension of the main result of P\`ene [Comm. Math. Phys. 225
(2002) 91-119] to the case of d-dimensional functions. In a second time, we use
the same method to establish a result of convergence in distribution for
(\frac{e_t^{\epsilon}(x,\cdot)}{\sqrt\epsilon})_{\epsilon>0} with rate of
convergence in O(\epsilon^{1/2-\alpha}) (for the Prokhorov metric).
http://arXiv.org/abs/math/0509008
http://front.math.ucdavis.edu/math.PR/0509008
(alternate)
Author(s): Thomas Duquesne (Paris 11)
Abstract: Let $b$ be an integer greater than 1 and let $W^{\ee}=(W^{\ee}_n; n\geq 0)$
be a random walk on the $b$-ary rooted tree $\U_b$, starting at the root, going
up (resp. down) with probability $1/2+\epsilon$ (resp. $1/2 -\epsilon$),
$\epsilon \in (0, 1/2)$, and choosing direction $i\in \{1, ..., b\}$ when going
up with probability $a_i$. Here $\aa =(a_1, ..., a_b)$ stands for some
non-degenerated fixed set of weights. We consider the range $\{W^{\ee}_n ;
n\geq 0 \}$ that is a subtree of $\U_b $. It corresponds to a unique random
rooted ordered tree that we denote by $\tau_{\epsilon}$. We rescale the edges
of $\tau_{\epsilon}$ by a factor $\ee $ and we let $\ee$ go to 0: we prove that
correlations due to frequent backtracking of the random walk only give rise to
a deterministic phenomenon taken into account by a positive factor $\gamma
(\aa)$. More precisely, we prove that $\tau_{\epsilon}$ converges to a
continuum random tree encoded by two independent Brownian motions with drift
conditioned to stay positive and scaled in time by $\gamma (\aa)$. We actually
state the result in the more general case of a random walk on a tree with an
infinite number of branches at each node ($b=\infty$) and for a general set of
weights $\aa =(a_n, n\geq 0)$.
http://arXiv.org/abs/math/0509524
http://front.math.ucdavis.edu/math.PR/0509524
(alternate)
Author(s): Jan Obloj (PMA and Mimuw)
Abstract: We develop an explicit non-randomized solution to the Skorokhod embedding
problem in an abstract setup of signed functionals of Markovian excursions. Our
setting allows to solve the Skorokhod embedding problem, in particular, for
diffusions and their (signed, scaled) age processes, for Azema's martingale,
for spectrally one-sided Levy processes and their reflected versions, for
Bessel processes of dimension smaller than 2, and for their age processes, as
well as for the age process of excursions of Cox-Ingersoll-Ross processes. This
work is a continuation and an important generalization of Obloj and Yor (SPA
110) [35]. Our methodology, following [35], is based on excursion theory and
the solution to the Skorokhod embedding problem is described in terms of the
Ito measure of the functional. We also derive an embedding for positive
functionals and we correct a mistake in the formula in [35] for measures with
atoms.
http://arXiv.org/abs/math/0509553
http://front.math.ucdavis.edu/math.PR/0509553
(alternate)
Author(s): Tom Schmitz
Abstract: This article is accepted for publication in the "Annals I.H.P. Prob. &
Stat.". We investigate the ballistic behavior of diffusions in random
environment. We introduce conditions in the spirit of (T) and (T') of the
discrete setting, cf. Sznitman \cite{szn01}, \cite{szn02}, that imply in higher
dimensions a strong law of large numbers with non-vanishing limiting velocity
(which we refer to as 'ballistic behavior') and a functional central limit
theorem with non-degenerate covariance matrix. As an application of our
results, we consider the class of diffusions where the diffusion matrix is the
identity, and give a concrete criterion on the drift term under which the
diffusion in random environment exhibits ballistic behavior. This criterion
provides new examples of ballistic diffusions in random environment.
http://arXiv.org/abs/math/0509554
http://front.math.ucdavis.edu/math.PR/0509554
(alternate)
Author(s): Thomas Duquesne (Paris 11) and Jean-Francois Le Gall (Ecole Normale Superieure de Paris and Paris 6)
Abstract: We investigate the genealogical structure of general critical or subcritical
continuous-state branching processes. Analogously to the coding of a discrete
tree by its contour function, this genealogical structure is coded by a
real-valued stochastic process called the height process, which is itself
constructed as a local time functional of a Levy process with no negative
jumps. We present a detailed study of the height process and of an associated
measure-valued process called the exploration process, which plays a key role
in most applications. Under suitable assumptions, we prove that whenever a
sequence of rescaled Galton-Watson processes converges in distribution, their
genealogies also converge to the continuous branching structure coded by the
appropriate height process. We apply this invariance principle to various
asymptotics for Galton-Watson trees. We then use the duality properties of the
exploration process to compute explicitly the distribution of the reduced tree
associated with Poissonnian marks in the height process, and the
finite-dimensional marginals of the so-called stable continuous tree. This last
calculation generalizes to the stable case a result of Aldous for the Brownian
continuum random tree. Finally, we combine the genealogical structure with an
independent spatial motion to develop a new approach to superprocesses with a
general branching mechanism. In this setting, we derive certain explicit
distributions, such as the law of the spatial reduced tree in a domain,
consisting of the collection of all historical paths that hit the boundary.
http://arXiv.org/abs/math/0509558
http://front.math.ucdavis.edu/math.PR/0509558
(alternate)
Author(s): Giovanni Peccati (LSTA) and Jean-Renaud Pycke (DP)
Abstract: Let G be a topological compact group acting on some space Y. We study a
decomposition of Y-indexed stochastic processes, based on the orthogonality
relations between the characters of the irreducible representations of G. In
the particular case of a Gaussian process with a G-invariant law, such a
decomposition gives a very general explanation of a classic identity in law -
between quadratic functionals of a Brownian bridge - due to Watson (1961).
Several relations with Karhunen-Lo\`{e}ve expansions are discussed, and some
applications and extensions are given - in particular related to Gaussian
processes indexed by a torus.
http://arXiv.org/abs/math/0509569
http://front.math.ucdavis.edu/math.PR/0509569
(alternate)
Author(s): Constantinos Daskalakis and Elchanan Mossel and Sebastien Roch
Abstract: It is well known that in order to reconstruct a tree on $n$ leaves, sequences
of length $\Omega(\log n)$ are needed. It was conjectured by M. Steel that for
the CFN evolutionary model, if the mutation probability on all edges of the
tree is less than $p^{\ast} = (\sqrt{2}-1)/2^{3/2}$ than the tree can be
recovered from sequences of length $O(\log n)$. This was proven by the second
author in the special case where the tree is ``balanced''. The second author
also proved that if all edges have mutation probability larger than $p^{\ast}$
then the length needed is $n^{\Omega(1)}$. This ``phase-transition'' in the
number of samples needed is closely related to the phase transition for the
reconstruction problem (or extremality of free measure) studied extensively in
statistical physics and probability.
Here we complete the proof of Steel's conjecture and give a reconstruction
algorithm using optimal (up to a multiplicative constant) sequence length. Our
results further extend to obtain optimal reconstruction algorithm for the
Jukes-Cantor model with short edges. All reconstruction algorithms run in time
polynomial in the sequence length.
The algorithm and the proofs are based on a novel combination of
combinatorial, metric and probabilistic arguments.
http://arXiv.org/abs/math/0509575
http://front.math.ucdavis.edu/math.PR/0509575
(alternate)
Author(s): Elchanan Mossel and Sebastien Roch
Abstract: We consider a group of agents on a graph who repeatedly play the prisoner's
dilemma game against their neighbors. The players adapt their actions to the
past behavior of their opponents by applying the win-stay lose-shift strategy.
On a finite connected graph, it is easy to see that the system learns to
cooperate by converging to the all-cooperate state in a finite time. We analyze
the rate of convergence in terms of the size and structure of the graph. [Dyer
et al., 2002] showed that the system converges rapidly on the cycle, but that
it takes a time exponential in the size of the graph to converge to cooperation
on the complete graph. We show that the emergence of cooperation is
exponentially slow in some expander graphs. More surprisingly, we show that it
is also exponentially slow in bounded-degree trees, where many other dynamics
are known to converge rapidly.
http://arXiv.org/abs/math/0509576
http://front.math.ucdavis.edu/math.PR/0509576
(alternate)
Author(s): Aniello Fedullo and Vitalii A. Gasanenko
Abstract: We have random number of independent diffusion processes with absorption on
boundaries in some region at initial time $t=0$. The initial numbers and
positions of processes in region is defined by Poisson random measure. It is
required to estimate of number of the unabsorbed processes for the fixed time
\~$\tau>0$. The Poisson random measure depends on $\tau$ and $\tau\to\infty$.
http://arXiv.org/abs/math/0509585
http://front.math.ucdavis.edu/math.PR/0509585
(alternate)
Author(s): Vitalii A. Gasanenko
Abstract: This paper deals with study of the sufficient condition of approximation
raring process with mixing by renewall process. We consider use the proved
results to practice problem too
http://arXiv.org/abs/math/0509586
http://front.math.ucdavis.edu/math.PR/0509586
(alternate)
Author(s): Serguei Foss and Takis Konstantopoulos and Stan Zachary
Abstract: We consider a modulated process S which, conditional on a background process
X, has independent increments. Assuming that S drifts to -infinity and that its
increments (jumps) are heavy-tailed (in a sense made precise in the paper), we
exhibit natural conditions under which the asymptotics of the tail distribution
of the overall maximum of S can be computed. We present results in discrete and
in continuous time. In particular, in the absence of modulation, the process S
in continuous time reduces to a Levy process with heavy-tailed Levy measure. A
central point of the paper is that we make full use of the so-called
``principle of a single big jump'' in order to obtain both upper and lower
bounds. Thus, the proofs are entirely probabilistic. The paper is motivated by
queueing and Levy stochastic networks.
http://arXiv.org/abs/math/0509605
http://front.math.ucdavis.edu/math.PR/0509605
(alternate)
Author(s): Catherine Donati-Martin (PMA) and Marc Yor (PMA)
Abstract: In a previous paper, we have shown that the gamma subordinators may be
represented as inverse local times of certain diffusions. In the present paper,
we give such representations for other subordinators whose L\'evy densities are
of the form $ \frac{\mathcal{C}}{(\sinh(y))^\gamma}$, $0 < \gamma < 2$, and the
more general family obtained from those by exponential tilting.
http://arXiv.org/abs/math/0509041
http://front.math.ucdavis.edu/math.PR/0509041
(alternate)
Author(s): Donald I. Cartwright and Wolfgang Woess
Abstract: A network is a countable, connected graph X viewed as a one-complex, where
each edge [x,y]=[y,x] (x,y in X^0, the vertex set) is a copy of the unit
interval within the graph's one-skeleton X^1 and is assigned a positive
conductance c(xy). A reference "Lebesgue" measure on X^1 is built up by using
Lebesgue measure with total mass c(xy) on each edge [x,y]. There are three
natural operators on X : the transition operator P acting on functions on X^0
(the reversible Markov chain associated with the conductances), the averaging
operator A over spheres of radius 1 on X^1, and the Laplace operator on X^1
(with Kirchhoff conditions weighted by c(.) at the vertices). The relation
between the l^2-spectrum of P and the H^2-spectrum of the Laplacian was
described by Cattaneo (Mh. Math. 124, 1997). In this paper we describe the
relation between the l^2-spectrum of P and the L^2-spectrum of A.
http://arXiv.org/abs/math/0509595
http://front.math.ucdavis.edu/math.FA/0509595
(alternate)
Author(s): Martin Hutzenthaler and Anton Wakolbinger
Abstract: For a class of processes modeling the evolution of a spatially structured
population with migration and a logistic local regulation of the reproduction
dynamics we show convergence towards an upper invariant measure from a suitable
class of initial distributions. It follows from recent work of A. Etheridge
that this upper invariant measure is non-trivial for sufficiently large
super-criticality in the reproduction. For sufficiently small super-criticality
we prove local extinction by comparison with a mean field model. This latter
result extends also to more general local reproduction regulations.
http://arXiv.org/abs/math/0509612
http://front.math.ucdavis.edu/math.PR/0509612
(alternate)
Author(s): Bronislovas Kaulakys and Julius Ruseckas and Vygintas Gontis and Miglius Alaburda
Abstract: Starting from the developed generalized point process model of $1/f$ noise
(B. Kaulakys et al, Phys. Rev. E 71 (2005) 051105; cond-mat/0504025) we derive
the nonlinear stochastic differential equations for the signal exhibiting
1/f^{\beta}$ noise and $1/x^{\lambda}$ distribution density of the signal
intensity with different values of $\beta$ and $\lambda$. The processes with
$1/f^{\beta}$ are demonstrated by the numerical solution of the derived
equations with the appropriate restriction of the diffusion of the signal in
some finite interval. The proposed consideration may be used for modeling and
analysis of stochastic processes in different systems with the power-law
distributions, long-range memory or with the elements of self-organization.
http://arXiv.org/abs/cond-mat/0509626
http://front.math.ucdavis.edu/cond-mat/0509626
(alternate)
Author(s): Marc Kesseb\"ohmer and Mehdi Slassi
Abstract: We consider conservative ergodic measure preserving transformations on
infinite measure spaces and investigate the asymptotic behaviour of distorted
return time processes with respect to sets satisfying a type of Darling-Kac
condition. As applications we derive asymptotic laws for the normalized Kac
process and the normalized spent time Kac process. We introduce the notion of
uniformly returning sets, for which we prove that if the wandering rate is
slowly varying then the normalized spent time Kac process converges strongly
distributional to a random variable uniformly distributed on the unit interval.
http://arXiv.org/abs/math/0509609
http://front.math.ucdavis.edu/math.DS/0509609
(alternate)
Author(s): Mark Adler & Pierre van Moerbeke
Abstract: The present paper studies a Gaussian Hermitian random matrix ensemble with
external source, given by a fixed diagonal matrix with two eigenvalues a and
-a. As a first result, the probability that the eigenvalues of the ensemble
belong to a set satisfies a fourth order PDE with quartic non-linearity; the
variables being the eigenvalue a and the boundary points of the set. This
equation enables one to find a PDE for the Pearcey distribution. The latter
describes the statistics of the eigenvalues near the closure of a gap; i.e.,
when the support of the equilibrium measure for large size random matrices has
a gap, which can be made to close. Precisely, the Gaussian Hermitian random
matrix ensemble with external source has this feature. In this work, we show
the Pearcey distribution satisfies a a fourth order PDE with cubic
non-linearity. The PDE for the finite problem is found by by showing that an
appropriate integrable deformation of the random matrix ensemble with external
source satisfies the three-component KP equation and Virasoro constraints.
http://arXiv.org/abs/math/0509047
http://front.math.ucdavis.edu/math.PR/0509047
(alternate)
Author(s): Magda Peligrad and Sergey Utev
Abstract: We establish the central limit theorem for linear processes with dependent
innovations including martingales and mixingale type of assumptions as defined
in McLeisch (1977) and motivated by Gordin (1969). In doing so we shall
preserve the generality of the coefficients, including the long range
dependence case, and we shall express the variance of partial sums in a form
easy to apply. Ergodicity is not required.
http://arXiv.org/abs/math/0509682
http://front.math.ucdavis.edu/math.PR/0509682
(alternate)
Author(s): Thomas Duquesne (Universite Paris 11); Jean-Francois Le Gall (Ecole Normale superieure et Universite Paris 6)
Abstract: We study fine properties of the so-called stable trees, which are the scaling
limits of critical Galton-Watson trees conditioned to be large. In particular
we derive the exact Hausdorff measure function for Aldous' continuum random
tree and for its level sets. It follows that both the uniform measure on the
tree and the local time measure on a level set coincide with certain Hausdorff
measures. Slightly less precise results are obtained for the Hausdorff measure
of general stable trees.
http://arXiv.org/abs/math/0509690
http://front.math.ucdavis.edu/math.PR/0509690
(alternate)
Author(s): Yvan Velenik (LMRS)
Abstract: The study of effective interface models has been quite active recently, with
a particular emphasis on the effect of various external potentials (wall,
pinning potential, ...) leading to localization/delocalization transitions. I
review some of the results that have been obtained. In particular, I discuss
pinning by a local potential, entropic repulsion and the (pre)wetting
transition, both for models with continuous and discrete heights. This text is
based on lecture notes for a mini-course given during the workshop "Topics in
Random Interfaces and Directed Polymers" held in Leipzig, September 12-17 2005.
http://arXiv.org/abs/math/0509695
http://front.math.ucdavis.edu/math.PR/0509695
(alternate)
Author(s): Huyen Pham (PMA)
Abstract: This paper is a survey on some recent aspects and developments in stochastic
control theory. We discuss the two main historical approaches, Bellman's
optimality principle and Pontryagin's maximum principle, and their modern
exposition with viscosity solutions and backward stochastic differential
equations. Some original proofs are presented in a unifying context including
degenerate singular controlControlled diffusions, dynamic programming, maximum
principle, viscosity solutions, backward stochastic differential equations,
finance. problems. We emphasize key results on characterization of optimal
control for diffusion processes, with a view towards applications. Some
examples in finance are detailed with their explicit solutions. We also discuss
numerical issues and open questions.
http://arXiv.org/abs/math/0509711
http://front.math.ucdavis.edu/math.PR/0509711
(alternate)
Author(s): Fabien Panloup (PMA)
Abstract: We investigate some recursive procedures based on an exact or ``approximate''
Euler scheme with decreasing step in vue to computation of invariant measures
of solutions to S.D.E. driven by a L\'{e}vy process. Our results are valid for
a large class of S.D.E. that can be governed by L\'{e}vy processes with few
moments or can have a weakly mean-reverting drift, and permit to find again the
a.s. C.L.T for stable processes.
http://arXiv.org/abs/math/0509712
http://front.math.ucdavis.edu/math.PR/0509712
(alternate)
Author(s): Jacky Cresson (LM-Besan\c{c}on) and S\'{e}bastien Darses (LM-Besan\c{c}on)
Abstract: Most physical systems are modelled by an ordinary or a partial differential
equation, like the n-body problem in celestial mechanics. In some cases, for
example when studying the long term behaviour of the solar system or for
complex systems, there exist elements which can influence the dynamics of the
system which are not well modelled or even known. One way to take these
problems into account consists of looking at the dynamics of the system on a
larger class of objects, that are eventually stochastic. In this paper, we
develop a theory for the stochastic embedding of ordinary differential
equations. We apply this method to Lagrangian systems. In this particular case,
we extend many results of classical mechanics namely, the least action
principle, the Euler-Lagrange equations, and Noether's theorem. We also obtain
a Hamiltonian formulation for our stochastic Lagrangian systems. Many
applications are discussed at the end of the paper.
http://arXiv.org/abs/math/0509713
http://front.math.ucdavis.edu/math.PR/0509713
(alternate)
Author(s): Jon Warren
Abstract: A family of reflected Brownian motions is used to construct Dyson's process
of non-colliding Brownian motions. A number of explicit formulae are given,
including one for the distribution of a family of coalescing Brownian motions.
http://arXiv.org/abs/math/0509720
http://front.math.ucdavis.edu/math.PR/0509720
(alternate)
Author(s): Amine Asselah Fabienne Castell
Abstract: We consider Random Walk in Random Scenery , denoted $X_n$, where the random
walk is symmetric on $Z^d$, with $d>4$, and the random field is made up of
i.i.d random variables with a stretched exponential tail decay, with exponent
$\alpha$ with $1<\alpha$. We present asymptotics for the probability, over both
randomness, that $\{X_n>n^{\beta}\}$ for $1/2<\beta<1$. To obtain such
asymptotics, we establish large deviations estimates for the the
self-intersection local times process.
http://arXiv.org/abs/math/0509721
http://front.math.ucdavis.edu/math.PR/0509721
(alternate)
Author(s): Esteban Moro and Henri Schurz
Abstract: Construction of splitting-step methods and properties of related
non-negativity and boundary preserving numerical algorithms for solving
stochastic differential equations (SDEs) of Ito-type are discussed. We present
convergence proofs for a newly designed splitting-step algorithm and simulation
studies for numerous numerical examples ranging from stochastic dynamics
occurring in asset pricing theory in mathematical finance (SDEs of CIR and CEV
models) to measure-valued diffusion and superBrownian motion (SPDEs) as met in
biology and physics.
http://arXiv.org/abs/math/0509724
http://front.math.ucdavis.edu/math.NA/0509724
(alternate)
Author(s): Robert Sh. Liptser and Anatolii A. Pukhalskii
Abstract: We consider a sequence $X^n=(X^n_t)_{t\ge 0},n\ge 1$ of semimartingales. Each
$X^n$ is a weak solution to an It\^o equation with respect to a Wiener process
and a Poissonian martingale measure and is in general non-Markovian process.
For this sequence, we prove the large deviation principle in the Skorokhod
space $D=D_{[0,\infty)}$. We use a new approach based on of exponential
tightness. This allows us to establish the large deviation principle under
weaker assumptions than before.
http://arXiv.org/abs/math/0510028
http://front.math.ucdavis.edu/math.PR/0510028
(alternate)
Author(s): R. Liptser
Abstract: We formulate large deviations principle (LDP) for diffusion pair
$(X^\epsilon,\xi^\epsilon)=(X_t^\epsilon,\xi_t^\epsilon)$, where first
component has a small diffusion parameter while the second is ergodic Markovian
process with fast time. More exactly, the LDP is established for
$(X^\epsilon,\nu^\epsilon)$ with $\nu^\epsilon(dt,dz)$ being an occupation type
measure corresponding to $\xi_t^\epsilon$. In some sense we obtain a
combination of Freidlin-Wentzell's and Donsker-Varadhan's results. Our approach
relies the concept of the exponential tightness and Puhalskii's theorem.
http://arXiv.org/abs/math/0510029
http://front.math.ucdavis.edu/math.PR/0510029
(alternate)
Author(s): Jean B\'erard and Jean-Baptiste Gou\'er\'e and Didier Piau
Abstract: We prove that a wide class of models of Markov neighbor-dependent
substitution processes on the integer line is solvable. This class contains
some models of nucleotide substitutions recently introduced and studied
empirically by molecular biologists. We show that the frequency of every
polynucleotide at equilibrium solves an explicit finite-sized linear system.
Finally, the dynamics of the process and the distribution at equilibrium
exhibit some stringent, unexpected, independence properties. For example,
nucleotide sites at distance at least three evolve independently, and the
sites, if encoded as purines and pyrimidines, evolve independently.
http://arXiv.org/abs/math/0510034
http://front.math.ucdavis.edu/math.PR/0510034
(alternate)
Author(s): Didier Piau
Abstract: We study the harmonic moments of Galton-Watson processes, possibly non
homogeneous, with positive values. Good estimates of these are needed to
compute unbiased estimators for non canonical branching
Markov processes, which occur, for instance, in the modeling of the polymerase
chain reaction. By convexity, the ratio of the harmonic mean to the mean is at
most 1. We prove that, for every square integrable branching mechanisms, this
ratio lies between 1-A/k and 1-B/k for every initial population of size k
greater than A. The positive constants A and B, such that B is at most A, are
explicit and depend only on the generation-by-generation branching mechanisms.
In particular, we do not use the distribution of the limit of the classical
martingale associated to the Galton-Watson process. Thus, emphasis is put on
non asymptotic bounds and on the dependence of the harmonic mean upon the size
of the initial population. In the Bernoulli case, which is relevant for the
modeling of the polymerase chain reaction, we prove essentially optimal bounds
that are valid for every initial population. Finally, in the general case and
for large enough initial populations, similar techniques yield sharp estimates
of the harmonic moments of higher degrees.
http://arXiv.org/abs/math/0510035
http://front.math.ucdavis.edu/math.PR/0510035
(alternate)
Author(s): Didier Piau
Abstract: We study some stochastic models of physical mapping of genomic sequences. Our
starting point is a global construction of the process of the clones and of the
process of the anchors which are used to map the sequence. This yields explicit
formulas for the moments of the proportion occupied by the anchored clones,
even in inhomogeneous models. This also allows to compare, in this respect,
inhomogeneous models to homogeneous ones. Finally, for homogeneous models, we
provide nonasymptotic bounds of the variance and we prove functional invariance
results.
http://arXiv.org/abs/math/0510036
http://front.math.ucdavis.edu/math.PR/0510036
(alternate)
Author(s): Didier Piau
Abstract: We study the iterated Galton-Watson process (IGW), possibly with thinning,
introduced by Gawe{\l}and Kimmel to model the number of repeats of DNA triplets
during some genetic disorders. If the process involves some thinning, then
extinction and explosion can have positive probability simultaneously. If the
underlying (simple) Galton-Watson process is nondecreasing with mean m, then,
conditionally on the explosion, the logarithm of the population of the IGW at
time n+1 is equivalent to log(m) times the population at time n, almost surely.
This simplifies arguments of Gawe{\l}and Kimmel, and confirms and extends a
conjecture of Pakes.
http://arXiv.org/abs/math/0510037
http://front.math.ucdavis.edu/math.PR/0510037
(alternate)
Author(s): Didier Piau
Abstract: According to Comets, Gantert and Zeitouni on the one hand and to Derriennic
on the other hand, some functionals associated to the hitting times of random
walks in random environment on the integer line coincide, for the walk itself
and for the walk in the reversed environment. We show that these two duality
principles are algebraically equivalent, that they both stem from the Markov
property of the walk in a fixed environment, and not of the ergodicity of the
model, and that there exists finitist and almost sure versions of this duality.
http://arXiv.org/abs/math/0510038
http://front.math.ucdavis.edu/math.PR/0510038
(alternate)
Author(s): Alexander V. Gnedin
Abstract: Chain records is a new type of multidimensional record. We discuss how often
the chain records are broken when the background sampling is from the unit cube
with uniform distribution (or, more generally, from an arbitrary continuous
product distribution).
http://arXiv.org/abs/math/0510042
http://front.math.ucdavis.edu/math.PR/0510042
(alternate)
Author(s): Didier Piau
Abstract: Baum-Katz theorem asserts that the Cesaro means of i.i.d. increments
distributed like X r-converge if and only if |X|^{r+1} is integrable. We
generalize this, and we unify other results, by proving that the following
equivalence holds, if and only if G is moderate: the Cesaro means G-converge if
and only if G(L(a)) is integrable for every a if and only if |X|.G(|X|) is
integrable. Here, L(a) is the last time when the deviation of the Cesaro mean
from its limit exceeds a, and G-convergence is the analogue of r-convergence.
This solves a question about the asymptotic optimality of Wald's sequential
tests.
http://arXiv.org/abs/math/0510043
http://front.math.ucdavis.edu/math.PR/0510043
(alternate)
Author(s): Alexandre Rybko and Senya Shlosman and Alexandre Vladimirov
Abstract: We establish the averaging property for a queuing process with one server,
M(t)/GI/1. It is a new relation between the output flow rate and the input flow
rate, crucial in the study of the Poisson Hypothesis. Its implications include
the statement that the output flow always possesses more regularity than the
input flow.
http://arXiv.org/abs/math/0510046
http://front.math.ucdavis.edu/math.PR/0510046
(alternate)
Author(s): G. Giacomin (1) and F. L. Toninelli (2) ((1) Universite' de Paris 7 and (2) ENS Lyon, UMR--CNRS 5672)
Abstract: We analyze the localized phase of a general model of a directed polymer in
the proximity of an interface that separates two solvents. Each monomer unit
carries a charge, $\omega_n$, that determines the type (attractive or
repulsive) and the strength of its interaction with the solvents. In addition,
there is a polymer--interface interaction and we want to model the case in
which there are impurities $\tilde\omega_n$, that we call again charges, at the
interface. The charges are distributed in an in--homogeneous fashion along the
chain and at the interface: more precisely the model we consider is of quenched
disordered type.
It is well known that such a model undergoes a localization/delocalization
transition. We focus on the localized phase, where the polymer sticks to the
interface. Our new results include estimates on the exponential decay of
averaged correlations and the proof that the free energy is infinitely
differentiable away from the transition. Other results we prove, instead,
generalize earlier works that typically deal either with the case of copolymers
near an homogeneous interface ($\tilde\omega\equiv 0$) or with the case of
disordered pinning, where the only polymer--environment interaction is at the
interface ($\omega\equiv 0$). Moreover, with respect to most of the previous
literature, we work with rather general distributions of charges (we will
assume only a suitable concentration inequality).
http://arXiv.org/abs/math/0510047
http://front.math.ucdavis.edu/math.PR/0510047
(alternate)
Author(s): Vitalii A. Gasanenko
Abstract: We prove constructible sufficient conditions of lack of exit by solutions of
stochastic differential Ito's equations from domains with smooth boundaries
http://arXiv.org/abs/math/0510077
http://front.math.ucdavis.edu/math.PR/0510077
(alternate)
Author(s): Pascal Azerad (I3M) and Mohamed Mellouk (I3M)
Abstract: In this paper, we prove existence, uniqueness and regularity for a class of
stochastic partial differential equations with a fractional Laplacian driven by
a space-time white noise in dimension one. The equation we consider may also
include a reaction term.
http://arXiv.org/abs/math/0510107
http://front.math.ucdavis.edu/math.AP/0510107
(alternate)
Author(s): Antar Bandyopadhyay and Ofer Zeitouni
Abstract: We consider a model, introduced by Boldrighini, Minlos and Pellegrinotti, of
random walks in dynamical random environments on the integer lattice Z^d with
d>=1. In this model, the environment changes over time in a Markovian manner,
independently across sites, while the walker uses the environment at its
current location in order to make the next transition. In contrast with the
cluster expansions approach of Boldrighini, Minlos and Pellegrinotti, we follow
a probabilistic argument based on regeneration times. We prove an annealed SLLN
and invariance principle for any dimension, and provide a quenched invariance
principle for dimension d > 6, providing for d>6 an alternative to the
analytical approach of Boldrighini, Minlos and Pellegrinotti, with the added
benefit that it is valid under weaker assumptions. The quenched results use, in
addition to the regeneration times already mentioned, a technique introduced by
Bolthausen and Sznitman.
http://arXiv.org/abs/math/0509066
http://front.math.ucdavis.edu/math.PR/0509066
(alternate)
Author(s): D. Shlyakhtenko
Abstract: We prove a free probability analog of a result of
Artstein-Bally-Barthez-Naor. In particualar we prove that if X_{1},X_{2},...
are freely independent identically distributed random variables, then the free
entropy chi(X_{1}+...+X_{n}/\sqrt{n}) is monotone increasing for all n. Our
proof also leads to a slight simplification of the original argument in the
classical case.
http://arXiv.org/abs/math/0510103
http://front.math.ucdavis.edu/math.OA/0510103
(alternate)
Author(s): Marc Lelarge
Abstract: Tail asymptotics for the supremum of an independent subadditive process are
obtained as a function of the logarithmic moment generating function. We use
this analysis to obtain large deviations results for queueing networks in their
stationary regime. In the particular case of (max,plus)-linear recursions, the
rate of exponential decay of the stationary solution can be explicitly
computed.
http://arXiv.org/abs/math/0510117
http://front.math.ucdavis.edu/math.PR/0510117
(alternate)
Author(s): Noemi Kurt
Abstract: Consider the centered Gaussian field on the lattice $\mathbb{Z}^d,$ $d$ large
enough, with covariances given by the inverse of $\sum_{j=k}^K q_j(-\Delta)^j,$
where $\Delta$ is the discrete Laplacian and $\{q_j\}_{k\leq j\leq K}$ is a
polynomial satisfying certain additional conditions. We extend a previously
known result to show that the probability that all spins are nonnegative on a
box of side-length $N$ has an exponential decay at rate of order
$N^{d-2k}\log{N}.$ We are able to explicitly compute the constant, which is
given in terms of a higher-order capacity of the unit cube, analogous to the
known result for the lattice free field.
http://arXiv.org/abs/math/0510143
http://front.math.ucdavis.edu/math.PR/0510143
(alternate)
Author(s): Pawe{\l} J. Szab{\l}owki
Abstract: We present some properties of measures orthogonalizing set of q-Hermite
polynomials so called $q$-Gaussian measures. We also present an algorithm
simmulating i.i.d. sequencs of random variables having $q$-Gaussian
distribution.
http://arXiv.org/abs/math/0510153
http://front.math.ucdavis.edu/math.PR/0510153
(alternate)
Author(s): Cecile Monthus and Thomas Garel
Abstract: According to recent progresses in the finite size scaling theory of
disordered systems, thermodynamic observables are not self-averaging at
critical points when the disorder is relevant in the Harris criterion sense.
This lack of self-averageness at criticality is directly related to the
distribution of pseudo-critical temperatures $T_c(i,L)$ over the ensemble of
samples $(i)$ of size $L$. In this paper, we apply this analysis to disordered
Poland-Scheraga models with different loop exponents $c$,corresponding to
marginal and relevant disorder. In all cases, we numerically obtain a Gaussian
histogram of pseudo-critical temperatures $T_c(i,L)$ with mean $T_c^{av}(L)$
and width $\Delta T_c(L)$. For the marginal case $c=1.5$ corresponding to
two-dimensional wetting, both the width $\Delta T_c(L)$ and the shift
$[T_c(\infty)-T_c^{av}(L)]$ decay as $L^{-1/2}$, so the exponent is unchanged
($\nu_{random}=2=\nu_{pure}$) but disorder is relevant and leads to non
self-averaging at criticality. For relevant disorder $c=1.75$, the width
$\Delta T_c(L)$ and the shift $[T_c(\infty)-T_c^{av}(L)]$ decay with the same
new exponent $L^{-1/\nu_{random}}$ (where $\nu_{random} \sim 2.7 > 2 >
\nu_{pure}$) and there is again no self-averaging at criticality. Finally for
the value $c=2.15$, of interest in the context of DNA denaturation, the
transition is first-order in the pure case. In the presence of disorder, the
width $\Delta T_c(L) \sim L^{-1/2}$ dominates over the shift
$[T_c(\infty)-T_c^{av}(L)] \sim L^{-1}$, i.e. there are two correlation length
exponents $\nu=2$ and $\tilde \nu=1$ that govern respectively the
averaged/typical loop distribution.
http://arXiv.org/abs/cond-mat/0509479
http://front.math.ucdavis.edu/cond-mat/0509479
(alternate)
Author(s): Svante Janson and Wojciech Szpankowski
Abstract: Andersson and Nilsson introduced in 1993 a level-compressed trie (in short:
LC trie) in which a full subtree of a node is compressed to a single node of
degree being the size of the subtree. Recent experimental results indicated a
'dramatic improvement' when full subtrees are replaced by partially filled
subtrees. In this paper, we provide a theoretical justification of these
experimental results showing, among others, a rather moderate improvement of
the search time over the original LC tries. For such an analysis, we assume
that n strings are generated independently by a binary memoryless source with p
denoting the probability of emitting a 1. We first prove that the so called
alpha-fillup level (i.e., the largest level in a trie with alpha fraction of
nodes present at this level) is concentrated on two values with high
probability. We give these values explicitly up to O(1), and observe that the
value of alpha (strictly between 0 and 1) does not affect the leading term.
This result directly yields the typical depth (search time) in the alpha-LC
tries with p not equal to 1/2, which turns out to be C loglog n for an
explicitly given constant C (depending on p but not on alpha). This should be
compared with recently found typical depth in the original LC tries which is C'
loglog n for a larger constant C'. The search time in alpha-LC tries is thus
smaller but of the same order as in the original LC tries.
http://arXiv.org/abs/cs/0510017
http://front.math.ucdavis.edu/cs.DS/0510017
(alternate)
Author(s): Kurt Johansson
Abstract: The Tracy-Widom distribution that has been much studied in recent years can
be thought of as an extreme value distribution. We discuss interpolation
between the classical extreme value distribution $\exp(-\exp(-x))$, the Gumbel
distribution and the Tracy-Widom distribution. There is a family of
determinantal processes whose edge behaviour interpolates between a Poisson
process with density $\exp(-x)$ and the Airy kernel point process. This process
can be obtained as a scaling limit of a grand canonical version of a random
matrix model introduced by Moshe, Neuberger and Shapiro. We also consider the
deformed GUE ensemble, $M=M_0+\sqrt{2S} V$, with $M_0$ diagobal with
independent elements and $V$ from GUE. Here we do not see a transition from
Tracy-Widom to Gumbel, but rather a transition from Tracy-Widom to Gaussian.
http://arXiv.org/abs/math/0510181
http://front.math.ucdavis.edu/math.PR/0510181
(alternate)
Author(s): Amine Asselah (LATP) and Fabienne Castell (LATP)
Abstract: We consider Random Walk in Random Scenery, denoted $X\_n$, where the random
walk is symmetric on $Z^d$, with $d>4$, and the random field is made up of
i.i.d random variables with a stretched exponential tail decay, with exponent
$\alpha$ with $1<\alpha$. We present asymptotics for the probability, over both
randomness, that $\{X\_n>n^{\beta}\}$ for $1/2<\beta<1$. To obtain such
asymptotics, we establish large deviations estimates for the the
self-intersection local times process.
http://arXiv.org/abs/math/0510190
http://front.math.ucdavis.edu/math.PR/0510190
(alternate)
Author(s): Kurt Johansson
Abstract: We survey recent results on determinantal processes, random growth, random
tilings and their relation to random matrix theory.
http://arXiv.org/abs/math-ph/0510038
http://front.math.ucdavis.edu/math-ph/0510038
(alternate)
Author(s): V.P.Belavkin
Abstract: A brief presentation of the basic concepts in quantum probability theory is
given in comparison to the classical one. The notion of quantum white noise,
its explicit representation in Fock space, and necessary results of
noncommutative stochastic analysis and integration are outlined. Algebraic
differential equations that unify the quantum non Markovian diffusion with
continuous non demolition observation are derived. A stochastic equation of
quantum diffusion filtering generalising the classical Markov filtering
equation to the quantum flows over arbitrary *-algebra is obtained. A Gaussian
quantum diffusion with one dimensional continuous observation is considered.The
a posteriori quantum state difusion in this case is reduced to a linear quantum
stochastic filter equation of Kalman-Bucy type and to the operator Riccati
equation for quantum correlations. An example of continuous nondemolition
observation of the coordinate of a free quantum particle is considered,
describing a continuous collase to the stationary solution of the linear
quantum filtering problem found in the paper.
http://arXiv.org/abs/quant-ph/0510028
http://front.math.ucdavis.edu/quant-ph/0510028
(alternate)
Author(s): Wlodzimierz Bryc and Wojciech Matysiak and Jacek Wesolowski
Abstract: This paper is a continuation of our previous research on quadratic harnesses,
i.e. processes with linear regressions and quadratic conditional variances. In
this paper we define the class of orthogonal polynomials that is a
two-parameter extension of the Al-Salam--Chihara polynomials, we derive a
relation between these polynomials for different values of parameters, and we
use the relation to construct a new class of quadratic harnesses. A special
case of our construction is a simple transformation of a linear pure-birth
process with immigration followed by a linear pure death process.
http://arXiv.org/abs/math/0510208
http://front.math.ucdavis.edu/math.PR/0510208
(alternate)
Author(s): Christophe Sabot (UMPA-ENSL)
Abstract: The aim of this text is to establish some relations between Markov chains in
Dirichlet Environments on directed graphs and certain hypergeometric integrals
associated with a particular arrangement of hyperplanes. We deduce from these
relations and the computation of the connexion obtained by moving one
hyperplane of the arrangement some new relations on important functionals of
the Markov chain.
http://arXiv.org/abs/math/0510236
http://front.math.ucdavis.edu/math.PR/0510236
(alternate)
Author(s): J. Ben Hough
Abstract: The "hole probability" that the zero set of the time dependent planar
Gaussian analytic function f(z,t) = sum_(n=0)^infty a_n(t) z^n/sqrt(n!), where
a_n(t) are i.i.d. complex valued Ornstein-Uhlenbeck processes, does not
intersect a disk of radius R for all 0
http://arXiv.org/abs/math/0510237
http://front.math.ucdavis.edu/math.PR/0510237
(alternate)
Author(s): Larry Goldstein and Gesine Reinert
Abstract: A new class of distributional transformations is introduced, characterized by
equations relating function weighted expectations of test functions on a given
distribution to expectations of the transformed distribution on the test
function's higher order derivatives. The class includes the size and zero bias
transformations, and when specializing to weighting by polynomial functions,
relates distributional families closed under independent addition, and in
particular the infinitely divisible distributions, to the family of
transformations induced by their associated orthogonal polynomial systems. For
these families, generalizing a well known property of size biasing, sums of
independent variables are transformed by replacing summands chosen according to
a multivariate distribution on its index set by independent variables whose
distributions are transformed by members of that same family. A variety of the
transformations associated with the classical orthogonal polynomial systems
have as fixed points the original distribution, or a member of the same family
with different parameter.
http://arXiv.org/abs/math/0510240
http://front.math.ucdavis.edu/math.PR/0510240
(alternate)
Author(s): David Assaf and Larry Goldstein and Ester Samuel-Cahn
Abstract: Let $X_n,...,X_1$ be i.i.d. random variables with distribution function $F$.
A statistician, knowing $F$, observes the $X$ values sequentially and is given
two chances to choose $X$'s using stopping rules. The statistician's goal is to
stop at a value of $X$ as small as possible. Let $V_n^2$ equal the expectation
of the smaller of the two values chosen by the statistician when proceeding
optimally. We obtain the asymptotic behavior of the sequence $V_n^2$ for a
large class of $F$'s belonging to the domain of attraction (for the minimum)
${\cal D}(G^\alpha)$, where $G^\alpha(x)=[1-\exp(-x^\alpha)]{\bf I}(x \ge 0)$.
The results are compared with those for the asymptotic behavior of the
classical one choice value sequence $V_n^1$, as well as with the ``prophet
value" sequence $V_n^p=E(\min\{X_n,...,X_1\})$.
http://arXiv.org/abs/math/0510242
http://front.math.ucdavis.edu/math.PR/0510242
(alternate)
Author(s): Patrick Cattiaux (MODAL'X and CMAP) and Nathael Gozlan (MODAL'X)
Abstract: The aim of this paper is to use non asymptotic bounds for the probability of
rare events in the Sanov theorem, in order to study the asymptotics in
conditional limit theorems (Gibbs conditioning principle for thin sets).
Applications to stochastic mechanics or calibration problems for diffusion
processes are discussed.
http://arXiv.org/abs/math/0510257
http://front.math.ucdavis.edu/math.PR/0510257
(alternate)
Author(s): Patrick Cattiaux (MODAL'X and CMAP)
Abstract: $\mu$ being a nonnegative measure satisfying some log-Sobolev inequality, we
give conditions on F for the measure $\nu=e^{-2F} \mu$ to also satisfy some
log-Sobolev inequality. Explicit examples are studied.
http://arXiv.org/abs/math/0510258
http://front.math.ucdavis.edu/math.PR/0510258
(alternate)
Author(s): T. Bodineau and B. Derrida
Abstract: We study the large deviation functional of the current for the Weakly
Asymmetric Simple Exclusion Process in contact with two reservoirs.We compare
this functional in the large drift limit to the one of the Totally Asymmetric
Simple Exclusion Process, in particular to the Jensen-Varadhan functional.
Conjectures for generalizing the Jensen-Varadhan functional to open systems are
also stated.
http://arXiv.org/abs/cond-mat/0509179
http://front.math.ucdavis.edu/cond-mat/0509179
(alternate)
Author(s): Serguei Foss and Dmitry Korshunov
Abstract: Suppose $F$ is a distribution on the half-line $[0,\infty)$. We study the
limits of the ratios of tails $\bar{F*F}(x)/\bar F(x)$ as $x\to\infty$. We also
discuss the classes of distributions ${\mathcal S}$, ${\mathcal S}(\gamma)$,
and ${\mathcal S}^*$.
http://arXiv.org/abs/math/0510273
http://front.math.ucdavis.edu/math.PR/0510273
(alternate)
Author(s): A.V. Gnedin and Yu. Yakubovich
Abstract: A class of random discrete distributions $P$ is introduced by means of a
recursive splitting of unity. Assuming supercritical branching, we show that
for partitions induced by sampling from such $P$ a power growth of the number
of blocks is typical. Some known and some new partition structures appear when
$P$ is induced by a Dirichlet splitting.
http://arXiv.org/abs/math/0510305
http://front.math.ucdavis.edu/math.PR/0510305
(alternate)
Author(s): Damien Lamberton (LAMA) and Gilles Pag\`{e}s (PMA)
Abstract: In this paper we investigate the rate of convergence of the so-called
two-armed bandit algorithm in a financial context of asset allocation. The
behaviour of the algorithm turns out to be highly non-standard: no CLT whatever
the time scale, possible existence of two rate regimes.
http://arXiv.org/abs/math/0510351
http://front.math.ucdavis.edu/math.PR/0510351
(alternate)
Author(s): A. Fey-den Boer and F. Redig
Abstract: We define stabilizability of an infinite volume height configuration and of a
probability measure on height configurations. We show that for high enough
densities, a probability measure cannot be stabilized. We also show that in
some sense the thermodynamic limit of the uniform measures on the recurrent
configurations of the abelian sandpile model (ASM) is a maximal element of the
set of stabilizable measures. In that sense the self-organized critical
behavior of the ASM can be understood in terms of an ordinary transition
between stabilizable and non-stabilizable
http://arXiv.org/abs/math-ph/0510060
http://front.math.ucdavis.edu/math-ph/0510060
(alternate)
Author(s): A.Vershik
Abstract: We consider generic i.e., forming an everywhere dense massive subset classes
of Markov operators in the space $L^2(X,\mu)$ with a finite continuous measure.
Since there is a canonical correspondence that associates with each Markov
operator a multivalued measure-preserving transformation (i.e., a
polymorphism), as well as a stationary Markov chain, we can also speak about
generic polymorphisms and generic Markov chains. The most important and
inexpected generic properties of Markov operators (or Markov chains or
polymorphisms) is nonmixing and totally nondeterministicity. It was not known
even existence of such Markov operators (the first example due to
M.Rozenblatt). We suppose that this class coinsided with the class of special
random perturbations of $K$-automorphisms. This theory is measure theoretic
counterpart of the theory of nonselfadjoint contractions and its application.
http://arXiv.org/abs/math/0510320
http://front.math.ucdavis.edu/math.FA/0510320
(alternate)
Author(s): Svante Janson
Abstract: Let X be a continuous random variable. We study the characteristic function
and moments of the integer-valued random variable obtained by rounding X+a to
the nearest smallest integer, where a is a constant. The results can be
regarded as exact versions of Sheppard's correction.
Rounded variables of this type often occur as subsequence limits of sequences
of integer-valued random variable. This leads to oscillatory terms in
asymptotics for these variables, something that often has been observed, for
example in the analysis of several algorithms. We give some examples, including
applications to tries, digital search trees and Patricia tries.
http://arXiv.org/abs/math/0509009
http://front.math.ucdavis.edu/math.PR/0509009
(alternate)
Author(s): Jon. Aaronson and Tom Meyerovitch
Abstract: We show that a dissipative, ergodic measure preserving transformation of a
sigma-finite, non-atomic measure space always has many non-proportional,
absolutely continuous, invariant measures and is ergodic with respect to each
one of these.
http://arXiv.org/abs/math/0509093
http://front.math.ucdavis.edu/math.DS/0509093
(alternate)
Author(s): Witold Bednorz
Abstract: We prove that whenever there exists a majroizing measure on the metric space,
then each process with bounded increments has necessarily bounded samples. This
is a strengthening of one the main results in Talagrand's paper "Sample
boundedness of stochastic processes under increment conditions".
http://arXiv.org/abs/math/0510373
http://front.math.ucdavis.edu/math.PR/0510373
(alternate)
Author(s): Damien Lamberton (LAMA) and Gilles Pag\`{e}s (PMA)
Abstract: We study a two armed-bandit algorithm with penalty. We show the convergence
of the algorithm and establish the rate of convergence. For some choices of the
parameters, we obtain a central limit theorem in which the limit distribution
is characterized as the unique stationary distribution of a discontinuous
Markov process.
http://arXiv.org/abs/math/0510384
http://front.math.ucdavis.edu/math.PR/0510384
(alternate)
Author(s): Davis S. Dean and Satya N. Majumdar
Abstract: We study analytically the late time statistics of the number of particles in
a growing tree model introduced by Aldous and Shields. In this model, a cluster
grows in continuous time on a binary Cayley tree, starting from the root, by
absorbing new particles at the empty perimeter sites at a rate proportional to
c^{-l} where c is a positive parameter and l is the distance of the perimeter
site from the root. For c=1, this model corresponds to random binary search
trees and for c=2 it corresponds to digital search trees in computer science.
By introducing a backward Fokker-Planck approach, we calculate the mean and the
variance of the number of particles at large times and show that the variance
undergoes a `phase transition' at a critical value c=sqrt{2}. While for
c>sqrt{2} the variance is proportional to the mean and the distribution is
normal, for c
http://arXiv.org/abs/cond-mat/0510429
http://front.math.ucdavis.edu/cond-mat/0510429
(alternate)
Author(s): F. Rassoul-Agha and T. Seppalainen
Abstract: We consider a ballistic random walk in an i.i.d. random environment that does
not allow retreating in a certain fixed direction. Homogenization and
regeneration techniques combine to prove a law of large numbers and an averaged
invariance principle. The assumptions are non-nestling and $1+\e$ (resp.\
$2+\e$) moments for the step of the walk uniformly in the environment, for the
law of large numbers (resp. invariance principles). We also investigate
invariance principles under fixed environments, and invariance principles for
the environment-dependent mean of the walk.
http://arXiv.org/abs/math/0510392
http://front.math.ucdavis.edu/math.PR/0510392
(alternate)
Author(s): Jinho Baik and Alexei Borodin and Percy Deift and Toufic Suidan
Abstract: The bus transportation system in Cuernevaca, Mexico, has certain
distinguished, innovative features and has been the subject of an intriguing,
recent study by M. Krbalek and P. Seba. Krbalek and Seba analyzed the
statistics of bus arrivals on Line 4 close to the city center. They studied, in
particular, the bus spacing distribution and also the bus number variance
measuring the fluctuations of the total number of buses arriving at a fixed
location during a time interval T. Quite remarkably, it was found that these
two statistics are well modeled by the Gaussian Unitary Ensemble (GUE) of
random matrix theory. Our goal in this paper is to provide a plausible
explanation of these observations, and to this end we introduce a microscopic
model for the bus line that leads simply and directly to GUE.
http://arXiv.org/abs/math/0510414
http://front.math.ucdavis.edu/math.PR/0510414
(alternate)
Author(s): Roberto Oliveira and Joel Spencer
Abstract: Consider a balls-in-bins process in which each new ball goes into a given bin
with probability proportional to $f(n)$, where $n$ is the number of balls
currently in the bin and $f$ is a fixed positive function. It is known that
these so-called {\em balls-in-bins processes with feedback} have a monopolistic
regime: if $f(x)=x^p$ for $p>1$, then there is a finite time after which one of
the bins will receive all incoming balls.
Our goal in this paper is to quantify the onset of monopoly. We show that the
initial number of balls is large and bin 1 starts with a fraction $\alpha>1/2$
of the balls, then with very high probability its share of the total number of
balls never decreases significantly below $\alpha$. Thus a bin that obtains
more than half of the balls at a "large time" will most likely preserve its
position of leadership. However, the probability that the winning bin has a
non-negligible advantage after $n$ balls are in the system is
$\sim{const.}\times n^{1-p}$, and the number of balls in the losing bin has a
power-law tail. Similar results also hold for more general functions $f$.
http://arXiv.org/abs/math/0510415
http://front.math.ucdavis.edu/math.PR/0510415
(alternate)
Author(s): Anders Johansson and Anders \"Oberg and Mark Pollicott
Abstract: In this paper we present a new approach to studying g-measures which is based
upon local absolute continuity. We extend the result in [11] that square
summability of variations of g-functions ensures uniqueness of g-measures. The
first extension is to the case of countably many symbols. The second extension
is to some cases where $g \geq 0$, relaxing the earlier requirement in [11]
that inf g>0.
http://arXiv.org/abs/math/0509109
http://front.math.ucdavis.edu/math.DS/0509109
(alternate)
Author(s): Sourav Chatterjee
Abstract: We obtain an asymptotically sharp error bound in the classical
Sudakov-Fernique comparison inequality for finite collections of gaussian
random variables. Our proof is short and self-contained, and gives an easy
alternative argument for the classical inequality, extended to the case of
non-centered processes.
http://arXiv.org/abs/math/0510424
http://front.math.ucdavis.edu/math.PR/0510424
(alternate)
Author(s): H\'{e}l\`{e}ne Gu\'{e}rin (IRMAR) and Sylvie M\'{e}l\'{e}ard (MODAL'X) and Eulalia Nualart (LAGA)
Abstract: The aim of this paper is to obtain estimates for the density of the law of a
specific nonlinear diffusion process at any positive bounded time. This process
is issued from kinetic theory and is called Landau process, by analogy with the
associated deterministic Fokker-Planck-Landau equation. It is not Markovian,
its coefficients are not bounded and the diffusion matrix is degenerate.
Nevertheless, the specific form of the diffusion matrix and the nonlinearity
imply the non-degeneracy of the Malliavin matrix and then the existence and
smoothness of the density. In order to obtain a lower bound for the density,
the known results do not apply. However, our approach follows the main idea
consisting in discretizing the interval time and developing a recursive method.
To this aim, we prove and use refined results on conditional Malliavin
calculus. The lower bound implies the positivity of the solution of the Landau
equation, and partially answers to an analytical conjecture. We also obtain an
upper bound for the density, which again leads to an unusual estimate due to
the bad behavior of the coefficients.
http://arXiv.org/abs/math/0510439
http://front.math.ucdavis.edu/math.PR/0510439
(alternate)
Author(s): Roberto Oliveira and Joel Spencer
Abstract: We analyze an evolving network model of Krapivsky and Redner in which new
nodes arrive sequentially, each connecting to a previously existing node b with
probability proportional to the p-th power of the in-degree of b. We restrict
to the super-linear case p>1. When 1+1/k< p \leq 1 + 1/(k-1) the structure of
the final countable tree is determined. There is a finite tree T with
distinguished v (which has a limiting distribution) on which is "glued" a
specific infinite tree. v has an infinite number of children, an infinite
number of which have k-1 children, and there are only a finite number of nodes
(possibly only v) with k or more children. Our basic technique is to embed the
discrete process in a continuous time process using exponential random
variables, a technique that has previously been employed in the study of
balls-in-bins processes with feedback.
http://arXiv.org/abs/math/0510446
http://front.math.ucdavis.edu/math.PR/0510446
(alternate)
Author(s): Nicolas Champagnat (MODAL'X) and R\'{e}gis Ferri\`{e}re and Sylvie M\'{e}l\'{e}ard (MODAL'X)
Abstract: We are interested in modelling Darwinian evolution, resulting from the
interplay of phenotypic variation and natural selection through ecological
interactions. Our models are rooted in the microscopic, stochastic description
of a population of discrete individuals characterized by one or several
adaptive traits. The population is modelled as a stochastic point process whose
generator captures the probabilistic dynamics over continuous time of birth,
mutation, and death, as influenced by each individual's trait values, and
interactions between individuals. An offspring usually inherits the trait
values of her progenitor, except when a mutation causes the offspring to take
an instantaneous mutation step at birth to new trait values. We look for
tractable large population approximations. By combining various scalings on
population size, birth and death rates, mutation rate, mutation step, or time,
a single microscopic model is shown to lead to contrasting macroscopic limits,
of different nature: deterministic, in the form of ordinary, integro-, or
partial differential equations, or probabilistic, like stochastic partial
differential equations or superprocesses. In the limit of rare mutations, we
show that a possible approximation is a jump process, justifying rigorously the
so-called trait substitution sequence. We thus unify different points of view
concerning mutation-selection evolutionary models.
http://arXiv.org/abs/math/0510453
http://front.math.ucdavis.edu/math.PR/0510453
(alternate)
Author(s): Ivan Werner
Abstract: We continue development of the theory of Markov systems initiated in
\cite{Wer1}. In this paper, we introduce fundamental Markov systems associated
with random dynamical systems.
http://arXiv.org/abs/math/0509120
http://front.math.ucdavis.edu/math.PR/0509120
(alternate)
Author(s): Antar Bandyopadhyay and David Gamarnik
Abstract: We propose a new type of approximate counting algorithms for the problems of
enumerating the number of independent sets and proper colorings in low degree
graphs with large girth. Our algorithms are not based on a commonly used Markov
chain technique, but rather are inspired by developments in statistical physics
in connection with correlation decay properties of Gibbs measures and its
implications to uniqueness of Gibbs measures on infinite trees, reconstruction
problems and local weak convergence methods.
On a negative side, our algorithms provide $\epsilon$-approximations only to
the logarithms of the size of a feasible set (also known as free energy in
statistical physics). But on the positive side, our approach provides
deterministic as opposed to probabilistic guarantee on approximations.
Moreover, for some regular graphs we obtain explicit values for the counting
problem. For example, we show that every 4-regular $n$-node graph with large
girth has approximately $(1.494...)^n$ independent sets, and in every
$r$-regular graph with $n$ nodes and large girth the number of $q\geq
r+1$-proper colorings is approximately $[q(1-{1\over q})^{r\over 2}]^n$, for
large $n$. In statistical physics terminology, we compute explicitly the limit
of the log-partition function. We extend our results to random regular graphs.
Our explicit results would be hard to derive via the Markov chain method.
http://arXiv.org/abs/math/0510471
http://front.math.ucdavis.edu/math.PR/0510471
(alternate)
Author(s): Richard F. Bass and Krzysztof Burdzy
Abstract: We give a simple proof that in a Lipschitz domain in two dimensions with
Lipschitz constant one, there is pathwise uniqueness for the Skorokhod equation
governing reflecting Brownian motion.
http://arXiv.org/abs/math/0510473
http://front.math.ucdavis.edu/math.PR/0510473
(alternate)
Author(s): Djalil Chafai (LSProba and Umr181 Inra/Envt)
Abstract: This article provides entropic inequalities for binomial-Poisson
distributions, derived from the two points space. They describe in particular
the exponential dissipation of $\Phi$-entropies along the M/M/$\infty$ queue.
This simple queueing process appears as a model of "constant curvature", and
plays for the simple Poisson process the role played by the Ornstein-Uhlenbeck
process for Brownian Motion. These inequalities are exactly the local
inequalities of the M/M/$\infty$ process. Some of them are recovered by
semigroup interpolation. Additionally, we explore the behaviour of these
entropic inequalities under a particular scaling, which sees the
Ornstein-Uhlenbeck process as a fluid limit of M/M/$\infty$ queues. Proofs are
elementary and rely essentially on the development of a "$\Phi$-calculus".
http://arXiv.org/abs/math/0510488
http://front.math.ucdavis.edu/math.PR/0510488
(alternate)
Author(s): K. Hamza and F. C. Klebaner
Abstract: This note is concerned with an important for modelling question of existence
of solutions of stochastic partial differential equations as proper stochastic
processes, rather than processes in the generalized sense. We consider a first
order stochastic partial differential equations of the form $\pd Ut = DW$, and
$\pd Ut-\pd Ux= DW$, where $D$ is a differential operator and $W(t,x)$ is a
continuous but non-differentiable function (field).
We give a necessary and sufficient condition for stochastic equations to have
solutions as functions. The result is then applied to the equation for a yield
curve. Proofs are based on probability arguments.
http://arXiv.org/abs/math/0510495
http://front.math.ucdavis.edu/math.PR/0510495
(alternate)
Author(s): Marc Atlan (PMA) and H\'{e}lyette Geman (DRM) and Marc Yor (PMA)
Abstract: The rapidly growing hedge fund industry has provided individual and
institutional investors with new investment vehicles and styles of management.
It has also brought forward a new form of performance contract: hedge fund
managers receive incentive fees which are typically a fraction of the fund net
asset value (NAV) above its starting level - a rule known as high water mark.
Options on hedge funds are becoming increasingly popular, in particular because
they allow investors with limited capital to get exposure to this new asset
class. The goal of the paper is to propose a valuation of options on hedge
funds which accounts for the high water market rule. Mathematically, this
valuation will lead to an interesting use of local times of Brownian motion.
Option prices are numerically computed by inversion of their Laplace
transforms.
http://arXiv.org/abs/math/0510497
http://front.math.ucdavis.edu/math.PR/0510497
(alternate)
Author(s): Magda Peligrad and Sergey Utev
Abstract: Braverman, Mallows and Shepp (1995), showed that if the absolute moments of
partial sums of i.i.d. symmetric variables are equal to those of normal
variables, then the marginals have normal distribution. This fact suggested the
conjecture that probably the absolute moments alone characterize the
homogeneous process with independent increments. In this paper we prove a more
general result that gives a positive answer to this conjecture, and then apply
it in order to obtain the CLT for a class of dependent random variables under a
normalization involving the absolute moments of partial sums.
http://arXiv.org/abs/math/0510513
http://front.math.ucdavis.edu/math.PR/0510513
(alternate)
Author(s): Jianhong Shen
Abstract: In contemporary image and vision analysis, stochastic approaches demonstrate
great flexibility in representing and modeling complex phenomena, while
variational-PDE methods gain enormous computational advantages over Monte-Carlo
or other stochastic algorithms. In combination, the two can lead to much more
powerful novel models and efficient algorithms. In the current work, we propose
a stochastic-variational model for soft (or fuzzy) Mumford-Shah segmentation of
mixture image patterns. Unlike the classical hard Mumford-Shah segmentation,
the new model allows each pixel to belong to each image pattern with some
probability. We show that soft segmentation leads to hard segmentation, and
hence is more general. The modeling procedure, mathematical analysis, and
computational implementation of the new model are explored in detail, and
numerical examples of synthetic and natural images are presented.
http://arXiv.org/abs/math/0510485
http://front.math.ucdavis.edu/math.OC/0510485
(alternate)
Author(s): Davar Khoshnevisan
Abstract: We can view Brownian sheet as a sequence of interacting Brownian motions or
slices. Here we present a number of results about the slices of the sheet. A
common feature of our results is that they exhibit phase transition. In
addition, a number of open problems are presented.
http://arXiv.org/abs/math/0510518
http://front.math.ucdavis.edu/math.PR/0510518
(alternate)
Author(s): Geoffrey Grimmett
Abstract: The two-point correlation function of a Potts model on a graph $G$ may be
expressed in terms of the flow polynomials of `Poissonian' random graphs
derived from $G$ by replacing each edge by a Poisson-distributed number of
copies of itself. This fact extends to Potts models the so-called
random-current expansion of the Ising model.
http://arXiv.org/abs/math/0509127
http://front.math.ucdavis.edu/math.PR/0509127
(alternate)
Author(s): Gerard Ben Arous and Stanislav Molchanov and Alejandro F. Ramirez
Abstract: We describe a universal transition mechanism characterizing the passage to an
annealed behavior and to a regime where the fluctuations about this behavior
are Gaussian, for the long time asymptotics of the empirical average of the
expected value of the number of random walks which branch and annihilate on
${\mathbb Z}^d$, with stationary random rates. The random walks are
independent, continuous time rate $2d\kappa$, simple, symmetric, with $\kappa
\ge 0$. A random walk at $x\in{\mathbb Z}^d$, binary branches at rate $v_+(x)$,
and annihilates at rate $v_-(x)$. The random environment $w$ has coordinates
$w(x)=(v_-(x),v_+(x))$ which are i.i.d. We identify a natural way to describe
the annealed-Gaussian transition mechanism under mild conditions on the rates.
Indeed, we introduce the exponents
$F_\theta(t):=\frac{H_1((1+\theta)t)-(1+\theta)H_1(t)}{\theta}$, and assume
that $\frac{F_{2\theta}(t)-F_\theta(t)}{\theta\log(\kappa t+e)}\to\infty$ for
$|\theta|>0$ small enough, where $H_1(t):=\log < m(0,t)>$ and $$
denotes the average of the expected value of the number of particles $m(0,t,w)$
at time $t$ and an environment of rates $w$, given that initially there was
only one particle at 0. Then the empirical average of $m(x,t,w)$ over a box of
side $L(t)$ has different behaviors: if $ L(t)\ge e^{\frac{1}{d}
F_\epsilon(t)}$ for some $\epsilon >0$ and large enough $t$, a law of large
numbers is satisfied; if $ L(t)\ge e^{\frac{1}{d} F_\epsilon (2t)}$ for some
$\epsilon>0$ and large enough $t$, a CLT is satisfied. These statements are
violated if the reversed inequalities are satisfied for some negative
$\epsilon$. Applications to potentials with Weibull, Frechet and double
exponential tails are given.
http://arXiv.org/abs/math/0510519
http://front.math.ucdavis.edu/math.PR/0510519
(alternate)
Author(s): Francis Comets (PMA) and Vincent Vargas (PMA)
Abstract: In this note we give upper bounds for the free energy of discrete polymers in
random media. The bounds are given by the so-called generalized multiplicative
cascades from the statistical theory of turbulence. For the polymer model, we
derive that the quenched free energy is different from the annealed one in
dimension 1, for any finite temperature and general environment. This implies
localization of the polymer.
http://arXiv.org/abs/math/0510525
http://front.math.ucdavis.edu/math.PR/0510525
(alternate)
Author(s): Bernard Roynette (IEC) and Pierre Vallois (IEC) and Marc Yor (PMA)
Abstract: We determine the rate of decay of the expectation Z(t) of some multiplicative
functional related to Brownian motion up to time t. This permits to prove that
the Wiener measure, penalized by this multiplicative functional, converges as t
goes to infinity to a probability measure (p.m.) . We obtain the law of the
canonical process under this new p.m.
http://arXiv.org/abs/math/0510550
http://front.math.ucdavis.edu/math.PR/0510550
(alternate)
Author(s): Masanori Hino and Takashi Kumagai
Abstract: We consider a trace theorem for self-similar Dirichlet forms on self-similar
sets to self-similar subsets. In particular, we characterize the trace of the
domains of Dirichlet forms on the Sierpinski gaskets and the Sierpinski carpets
to their boundaries, where boundaries mean the triangles and rectangles which
confine gaskets and carpets. As an application, we construct diffusion
processes on a collection of fractals called fractal fields, which behave as
the appropriate fractal diffusion within each fractal component of the field.
http://arXiv.org/abs/math/0510553
http://front.math.ucdavis.edu/math.PR/0510553
(alternate)
Author(s): Nina Gantert and Sebastian Mueller
Abstract: We investigate recurrence and transience of Branching Markov Chains (BMC) in
discrete time. Branching Markov Chains are clouds of particles which move
(according to an irreducible underlying Markov Chain) and produce offspring
independently. The offspring distribution can depend on the location of the
particle. If the offspring distribution is constant for all locations, these
are Tree-Indexed Markov chains in the sense of \cite{benjamini94}. Starting
with one particle at location $x$, we denote by $\alpha(x)$ the probability
that $x$ is visited infinitely often by the cloud. Due to the irreducibility of
the underlying Markov Chain, there are three regimes: either $\alpha(x) = 0$
for all $x$ (transient regime), or $0 < \alpha(x) < 1$ for all $x$ (weakly
recurrent regime) or $\alpha(x) = 1$ for all $x$ (strongly recurrent regime).
We give classification results, including a sufficient condition for transience
in the general case.
If the mean of the offspring distribution is constant, we give a criterion for
transience involving the spectral radius of the underlying Markov Chain and the
mean of the offspring distribution.
http://arXiv.org/abs/math/0510556
http://front.math.ucdavis.edu/math.PR/0510556
(alternate)
Author(s): Alexander Gnedin and Denis Miretskiy
Abstract: Following a long-standing suggestion by Gilbert and Mosteller, we derive an
explicit formula for the asymptotic winning rate in the full-information
problem of the best choice.
http://arXiv.org/abs/math/0510568
http://front.math.ucdavis.edu/math.PR/0510568
(alternate)
Author(s): Bernard Roynette (IEC) and Pierre Vallois (IEC) and Marc Yor (PMA)
Abstract: We obtain probability measures on the canonical space penalizing the Wiener
measure by a function of its maximum (resp. minimum, local time). We study the
law of the canonical process under these new probability measures.
http://arXiv.org/abs/math/0510575
http://front.math.ucdavis.edu/math.PR/0510575
(alternate)
Author(s): Bela Bollobas and Ronald Meester and Oliver Riordan
Abstract: Recently, a short proof of the Harris-Kesten result that the critical
probability for bond percolation in the planar square lattice is 1/2 was given,
using a sharp threshold result of Friedgut and Kalai. Here we point out that a
key part of this proof may be replaced by an argument of Russo from 1982, using
his approximate zero-one law in place of the Friedgut-Kalai result. Russo's
paper gave a new proof of the Harris-Kesten Theorem that seems to have received
little attention.
http://arXiv.org/abs/math/0509131
http://front.math.ucdavis.edu/math.PR/0509131
(alternate)
Author(s): Larry Goldstein and Yosef Rinott
Abstract: Stein's method is used to obtain two theorems on multivariate normal
approximation. Our main theorem, Theorem 1.2, provides a bound on the distance
to normality for any nonnegative random vector. Theorem 1.2 requires
multivariate size bias coupling, which we discuss in studying the approximation
of distributions of sums of dependent random vectors. In the univariate case,
we briefly illustrate this approach for certain sums of nonlinear functions of
multivariate normal variables. As a second illustration, we show that the
multivariate distribution counting the number of vertices with given degrees in
certain random graphs is asymptotically multivariate normal and obtain a bound
on the rate of convergence. Both examples demonstrate that this approach may be
suitable for situations involving non-local dependence. We also present Theorem
1.4 for sums of vectors having a local type of dependence. We apply this
theorem to obtain a multivariate normal approximation for the distribution of
the random $p$-vector which counts the number of edges in a fixed graph both of
whose vertices have the same given color when each vertex is colored by one of
$p$ colors independently. All normal approximation results presented here do
not require an ordering of the summands related to the dependence structure.
This is in contrast to hypotheses of classical central limit theorems and
examples, which involve e.g., martingale, Markov chain, or various mixing
assumptions.
http://arXiv.org/abs/math/0510586
http://front.math.ucdavis.edu/math.PR/0510586
(alternate)
Author(s): David Assaf and Larry Goldstein and and Ester Samuel-Cahn
Abstract: A curious connection exists between the theory of optimal stopping for
independent random variables, and branching processes. In particular, for the
branching process $Z_n$ with offspring distribution $Y$, there exists a random
variable $X$ such that the probability $P(Z_n=0)$ of extinction of the $n$th
generation in the branching process equals the value obtained by optimally
stopping the sequence $X_1,...,X_n$, where these variables are i.i.d
distributed as $X$. Generalizations to the inhomogeneous and infinite horizon
cases are also considered. This correspondence furnishes a simple `stopping
rule' method for computing various characteristics of branching processes,
including rates of convergence of the $n^{th}$ generation's extinction
probability to the eventual extinction probability, for the supercritical,
critical and subcritical Galton-Watson process. Examples, bounds, further
generalizations and a connection to classical prophet inequalities are
presented. Throughout, the aim is to show how this unexpected connection can be
used to translate methods from one area of applied probability to another,
rather than to provide the most general results.
http://arXiv.org/abs/math/0510587
http://front.math.ucdavis.edu/math.PR/0510587
(alternate)
Author(s): Manjunath Krishnapur
Abstract: We consider the point process of zeroes of certain Gaussian analytic
functions and find the asymptotics for the probability that there are more than
m points of the process in a fixed disk of radius r, as m-->infinity. For the
Planar Gaussian analytic function, sum_n a_n z^n/sqrt(n!), we show that this
probability is asymptotic to exp(-0.5 m^2 log(m)). For the Hyperbolic Gaussian
analytic functions, sum_n sqrt({-rho choose n}) a_n z^n, rho>0, we show that
this probability decays like exp(-cm^2).
In the planar case, we also consider the problem posed by Mikhail Sodin on
moderate and very large deviations in a disk of radius r as r --> infinity. We
partly solve the problem by showing that there is a qualitative change in the
asymptotics of the probability as we move from the large deviation regime to
the moderate.
http://arXiv.org/abs/math/0510588
http://front.math.ucdavis.edu/math.PR/0510588
(alternate)
Author(s): Nathael Gozlan (MODAL'X) and Christian L\'{e}onard (MODAL'X and CMAP)
Abstract: New transportation cost inequalities are derived by means of elementary large
deviation reasonings. Their dual characterization is proved; this provides an
extension of a well-known result of S. Bobkov and F. G\"{o}tze. Their
tensorization properties are investigated. Sufficient conditions (and necessary
conditions too) for these inequalities are stated in terms of the integrability
of the reference measure. Applying these results leads to new deviation
results: concentration of measure and deviations of empirical processes.
http://arXiv.org/abs/math/0510601
http://front.math.ucdavis.edu/math.PR/0510601
(alternate)
Author(s): Tom Kennedy
Abstract: The scaling limit of the two-dimensional self-avoiding walk (SAW) is believed
to be given by the Schramm-Loewner evolution (SLE) with the parameter kappa
equal to 8/3. The scaling limit of the SAW has a natural parameterization and
SLE has a standard parameterization using the half-plane capacity. These two
parameterizations do not correspond with one another. To make the scaling limit
of the SAW and SLE agree as parameterized curves, we must reparameterize one of
them. We present Monte Carlo results that show that if we reparameterize the
SAW using the half-plane capacity, then it agrees well with SLE with its
standard parameterization. We then consider how to reparameterize SLE to make
it agree with the SAW with its natural parameterization. We argue using Monte
Carlo results that the so-called p-variation of the SLE curve with p=1/nu=4/3
provides a parameterization that corresponds to the natural parameterization of
the SAW.
http://arXiv.org/abs/math/0510604
http://front.math.ucdavis.edu/math.PR/0510604
(alternate)
Author(s): Leandro P. R. Pimentel
Abstract: In this paper we study first-passge percolation models on Delaunay
triangulations. We show a sufficient condition to ensure that the asymptotic
value of the rescaled first-passage time, called the time constant, is strictly
positive and derive some upper bounds for fluctuations. Our proofs are based on
renormalization ideas and on the method of bounded increments.
http://arXiv.org/abs/math/0510605
http://front.math.ucdavis.edu/math.PR/0510605
(alternate)
Author(s): Larry Goldstein and Gesine Reinert
Abstract: Let $W$ be a random variable with mean zero and variance $\sigma^2$. The
distribution of a variate $W^*$, satisfying $EWf(W)=\sigma ^2 Ef'(W^*)$ for
smooth functions $f$, exists uniquely and defines the zero bias transformation
on the distribution of $W$. The zero bias transformation shares many
interesting properties with the well known size bias transformation for
non-negative variables, but is applied to variables taking on both positive and
negative values. The transformation can also be defined on more general random
objects. The relation between the transformation and the expression
$wf'(w)-\sigma^2 f''(w)$ which appears in the Stein equation characterizing the
mean zero, variance $\sigma ^2$ normal $\sigma Z$ can be used to obtain bounds
on the difference $E\{h(W/\sigma)-h(Z)\}$ for smooth functions $h$ by
constructing the pair $(W,W^*)$ jointly on the same space. When $W$ is a sum of
$n$ not necessarily independent variates, under certain conditions which
include a vanishing third moment, bounds on this difference of the order $1/n$
for classes of smooth functions $h$ may be obtained. The technique is
illustrated by an application to simple random sampling.
http://arXiv.org/abs/math/0510619
http://front.math.ucdavis.edu/math.PR/0510619
(alternate)
Author(s): F. Camia and L. R. G. Fontes and C. M. Newman
Abstract: We analyze the geometry of scaling limits of near-critical 2D percolation,
i.e., for $p=p_c+\lambda\delta^{1/\nu}$, with $\nu=4/3$, as the lattice spacing
$\delta \to 0$. Our proposed framework extends previous analyses for $p=p_c$,
based on $SLE_6$. It combines the continuum nonsimple loop process describing
the full scaling limit at criticality with a Poissonian process for marking
double (touching) points of that (critical) loop process. The double points are
exactly the continuum limits of "macroscopically pivotal" lattice sites and the
marked ones are those that actually change state as $\lambda$ varies. This
structure is rich enough to yield a one-parameter family of near-critical loop
processes and their associated connectivity probabilities as well as related
processes describing, e.g., the scaling limit of 2D minimal spanning trees.
http://arXiv.org/abs/cond-mat/0510740
http://front.math.ucdavis.edu/cond-mat/0510740
(alternate)
Author(s): Jaime A. Londo\~no
Abstract: An extension of the idea of state tameness is presented in a dynamic
framework. The proposed model for financial markets is rich enough to provide
analytical tools that are mostly obtained in models that arise as the solution
of SDEs with deterministic coefficients. In the presented model the
augmentation by a shadow stock of the price evolution has a Markovian
character. As in a previous paper, the results obtained on valuation of
European contingent claims and American contingent claims do not require the
full range of the volatility matrix. Under some additional continuity
conditions, the conceptual framework provided by the model makes it possible to
regard the valuation of financial instruments of the European type as a
particular case of valuation of instruments of American type. This provides a
unifying framework for the problem of valuation of financial instruments.
http://arXiv.org/abs/math/0509139
http://front.math.ucdavis.edu/math.PR/0509139
(alternate)
Author(s): W. Strauss and N. D. Macheras and K. Musial
Abstract: We prove that if (X,\mathfrakA,P) is an arbitrary probability space with
countably generated \sigma-algebra \mathfrakA, (Y,\mathfrakB,Q) is an arbitrary
complete probability space with a lifting \rho and \hat R is a complete
probability measure on \mathfrakA \hat \otimes_R \mathfrakB determined by a
regular conditional probability {S_y:y\in Y} on \mathfrakA with respect to
\mathfrakB, then there exist a lifting \pi on (X\times Y,\mathfrakA \hat
\otimes_R \mathfrakB,\hat R) and liftings \sigma_y on (X,\hat \mathfrakA_y,\hat
S_y), y\in Y, such that, for every E\in\mathfrakA \hat \otimes_R \mathfrakB and
every y\in Y, [\pi(E)]^y=\sigma_y\bigl([\pi(E)]^y\bigr). Assuming the absolute
continuity of R with respect to P\otimes Q, we prove the existence of a regular
conditional probability {T_y:y\in Y} and liftings \varpi on (X\times
Y,\mathfrakA \hat \otimes_R \mathfrakB,\hat R), \rho' on (Y,\mathfrakB,\hat Q)
and \sigma_y on (X,\hat \mathfrakA_y,\hat S_y), y\in Y, such that, for every
E\in\mathfrakA \hat \otimes_R \mathfrakB and every y\in Y,
[\varpi(E)]^y=\sigma_y\bigl([\varpi(E)]^y\bigr) and \varpi(A\times
B)=\bigcup_{y\in\rho'(B)}\sigma_y(A)\times{y}\qquadif A\times
B\in\mathfrakA\times\mathfrakB. Both results are generalizations of Musia\l,
Strauss and Macheras [Fund. Math. 166 (2000) 281-303] to the case of measures
which are not necessarily products of marginal measures. We prove also that
liftings obtained in this paper always convert \hat R-measurable stochastic
processes into their \hat R-measurable modifications.
http://arXiv.org/abs/math/0509010
http://front.math.ucdavis.edu/math.PR/0509010
(alternate)
Author(s): Yuri Bakhtin and Jonathan C. Mattingly
Abstract: We explore Ito stochastic differential equations where the drift term
possibly depends on the infinite past. Assuming the existence of a Lyapunov
function, we prove the existence of a stationary solution assuming only minimal
continuity of the coefficients. Uniqueness of the stationary solution is proven
if the dependence on the past decays sufficiently fast. The results of this
paper are then applied to stochastically forced dissipative partial
differential equations such as the stochastic Navier-Stokes equation and
stochastic Ginsburg-Landau equation.
http://arXiv.org/abs/math/0509166
http://front.math.ucdavis.edu/math.PR/0509166
(alternate)
Author(s): Jeffrey C. Lagarias and K. Soundararajan
Abstract: We show that for most choices of an initial seed $x_0$, the sequence of the
first $N$ iterates of $x_0$ under the $3x+1$ map approximately satisfies
Benford's law.
http://arXiv.org/abs/math/0509175
http://front.math.ucdavis.edu/math.NT/0509175
(alternate)
Author(s): Jean Mairesse and Fr\'ed\'eric Math\'eus
Abstract: This paper is an appendix to the paper "Random walks on free products of
cyclic groups" by J.Mairesse and F.Math\'eus. It contains the details of the
computations and the proofs of the results concerning the examples treated
there.
http://arXiv.org/abs/math/0509208
http://front.math.ucdavis.edu/math.PR/0509208
(alternate)
Author(s): Jean Mairesse and Fr\'ed\'eric Math'eus
Abstract: Let G be a free product of a finite family of finite groups, with the set of
generators being formed by the union of the finite groups. We consider a
transient nearest-neighbour random walk on G. We give a new proof of the fact
that the harmonic measure is a special Markovian measure entirely determined by
a finite set of polynomial equations. We show that in several simple cases of
interest, the polynomial equations can be explicitely solved, to get closed
form formulas for the drift. The examples considered are the modular group
Z/2Z*Z/3Z, Z/3Z*Z/3Z, Z/kZ*Z/kZ, and the Hecke groups Z/2Z*Z/kZ. We also use
these various examples to study Vershik's notion of extremal generators, which
is based on the relation between the drift, the entropy, and the volume of the
group.
http://arXiv.org/abs/math/0509211
http://front.math.ucdavis.edu/math.PR/0509211
(alternate)
Author(s): Anna Karczewska
Abstract: In the paper stochastic Volterra equations of nonscalar type are studied
using resolvent approach. The aim of this note is to provide some results on
stochastic convolution and integral mild solutions to those Volterra equations.
The motivation of the paper comes from a model of aging viscoelastic materials.
http://arXiv.org/abs/math/0509012
http://front.math.ucdavis.edu/math.PR/0509012
(alternate)
Author(s): Gady Kozma
Abstract: Let G be a planar graph with polynomial growth and isoperimetric dimension
bigger than 1. Then the critical p for Bernoulli percolation on G satisfies
p<1.
http://arXiv.org/abs/math/0509235
http://front.math.ucdavis.edu/math.PR/0509235
(alternate)
Author(s): Eran Ofek
Abstract: Let d be a sufficiently large constant. A (n,d,c sqrt{d}) graph G is a d
regular graph over n vertices whose second largest eigenvalue (in absolute
value) is at most c sqrt{d}. For any 0 < p < 1, G_p is the graph induced by
retaining each edge of G with probability p. We show that for any p >
5c/sqrt{d} the graph G_p almost surely contains a unique giant component (a
connected component with linear number vertices). We further show that the
giant component of G_p almost surely has an edge expansion of at least 1/(log_2
n).
http://arXiv.org/abs/math/0509253
http://front.math.ucdavis.edu/math.PR/0509253
(alternate)
Author(s): Pierre Mathieu
Abstract: We extend the Carne-Varopoulos upper bound on the probability transitions of
a Markov chain to a certain class of non-reversible processes by introducing
the definition of a `centering measure'. In the case of random walks on a
group, we study the connections between different notions of centering.
http://arXiv.org/abs/math/0509257
http://front.math.ucdavis.edu/math.PR/0509257
(alternate)
Author(s): Frederic Dambreville (DGA/CEP/GIP/SRO)
Abstract: In this paper a conditional logic is defined and studied. This conditional
logic, DmBL, is constructed as close as possible to the Bayesian and is
unrestricted, that is one is able to use any operator without restriction. A
notion of logical independence is also defined within the logic itself. This
logic is shown to be non trivial and is not reduced to classical propositions.
A model is constructed for the logic. Completeness results are proved. It is
shown that any unconditioned probability can be extended to the whole logic
DmBL. The Bayesian is then recovered from the probabilistic DmBL. At last, it
is shown why DmBL is compliant with Lewis triviality.
http://arXiv.org/abs/math/0509248
http://front.math.ucdavis.edu/math.LO/0509248
(alternate)
Author(s): Markos A Katsoulakis and Petr Plechac and Alexandros Sopasakis
Abstract: In this paper we investigate the approximation properties of the
coarse-graining procedure applied to kinetic Monte Carlo simulations of lattice
stochastic dynamics. We provide both analytical and numerical evidence that the
hierarchy of the coarse models is built in a systematic way that allows for
error control in both transient and long-time simulations. We demonstrate that
the numerical accuracy of the CGMC algorithm as an approximation of stochastic
lattice spin flip dynamics is of order two in terms of the coarse-graining
ratio and that the natural small parameter is the coarse-graining ratio over
the range of particle/particle interactions. The error estimate is shown to
hold in the weak convergence sense. We employ the derived analytical results to
guide CGMC algorithms and we demonstrate a CPU speed-up in demanding
computational regimes that involve nucleation, phase transitions and
metastability.
http://arXiv.org/abs/math/0509228
http://front.math.ucdavis.edu/math.NA/0509228
(alternate)
Author(s): Narcisse Randrianantoanina
Abstract: We prove a weak-type (1,1) inequality involving conditioned square functions
of martingales in non-commutative $L^p$-spaces associated with finite von
Neumann algebras. As application, we determine the optimal orders for the best
constants in the non-commutative Burkholder/Rosenthal inequalities from Ann.
Proba. 31 (2003), 948-995. We also discuss BMO-norms of sums of non-commuting
order independent operators.
http://arXiv.org/abs/math/0509226
http://front.math.ucdavis.edu/math.OA/0509226
(alternate)
Author(s): Barbara Forster and Eva Luetkebohmert and Josef Teichmann
Abstract: Calculation of Greeks by Malliavin weights has proved to be a numerically
satisfactory procedure for usual Ito-diffusions. In this article we prove
existence of Malliavin weights for jump diffusions under H\"{o}rmander
conditions and hypotheses on the invertibility of the linkage operators. The
main result -- in the hypo-ellitpic case -- is the invertibility of the
covariance matrix, which enables -- by usual methods -- the construction of the
relevant Malliavin weights. The message is that in fairly general
jump-diffusion cases one should proceed such as in pure diffusion cases. In
contrast to Davis et al. we do not need any separability assumptions.
http://arXiv.org/abs/math/0509016
http://front.math.ucdavis.edu/math.PR/0509016
(alternate)
Author(s): Shankar Bhamidi and Steven N. Evans and Ron Peled and Peter Ralph
Abstract: Motivated by L\'evy's characterization of Brownian motion on the line, we
propose an analogue of Brownian motion that has as its state space an arbitrary
unbounded closed subset of the line: such a process will be a Feller-Dynkin
process that is a martingale, has the identity function as its quadratic
variation process, and is ``continuous'' in the sense that its sample paths
don't skip over points. We show that there is a unique such process and find
its generator. We then consider the special case where the state space is the
self-similar set $\{\pm q^k : k \in \Z\} \cup \{0\}$ for some $q>1$. Using the
scaling properties of the process, we represent the Laplace transforms of
various hitting times as certain continued fractions that appear in Ramanujan's
``lost'' notebook and evaluate these continued fractions in terms of
$q$-analogues of classical hypergeometric functions. The process has 0 as a
regular instantaneous point, and hence its sample paths can be decomposed into
a Poisson process of excursions from 0 using the associated continuous local
time. We find the entrance laws of the corresponding It\^o excursion measure
and the Laplace exponent of the inverse local time -- both again in terms of
basic hypergeometric functions -- and hence obtain explicit formulae for the
resolvent of the process.
http://arXiv.org/abs/math/0509270
http://front.math.ucdavis.edu/math.PR/0509270
(alternate)
Author(s): Marianne Akian and Stephane Gaubert and Vassili Kolokoltsov
Abstract: We generalise the Gartner-Ellis theorem of large deviations theory. Our
results allow us to derive large deviation type results in stochastic optimal
control from the convergence of generalised logarithmic moment generating
functions. They rely on the characterisation of the uniqueness of the solutions
of max-plus linear equations. We give an illustration for a simple investment
model, in which logarithmic moment generating functions represent
risk-sensitive values.
http://arXiv.org/abs/math/0509279
http://front.math.ucdavis.edu/math.PR/0509279
(alternate)
Author(s): Patrick Cheridito and H. Mete Soner and Nizar Touzi and Nicolas Victoir
Abstract: We introduce a class of second order backward stochastic differential
equations and show relations to fully non-linear parabolic PDEs. In particular,
we provide a stochastic representation result for solutions of such PDEs and
discuss Monte Carlo methods for their numerical treatment.
http://arXiv.org/abs/math/0509295
http://front.math.ucdavis.edu/math.PR/0509295
(alternate)
Author(s): Didier Dacunha-Castelle and Lisandro Ferm\'{\i}n
Abstract: We prove that a large set of long memory (LM) processes (including classical
LM processes and all processes whose spectral densities have a countable number
of singularities controlled by exponential functions) are obtained by an
aggregation procedure involving short memory (SM) processes whose spectral
densities are infinitely differentiable (C^{infty}). We show that the C^{infty}
class of spectral densities is the optimal class to get a general result for
disaggregation of LM processes in SM processes, in the sense that the result
given in C^{infty} class cannot be improved taking for instance analytic
functions instead of indefinitely derivable functions.
http://arXiv.org/abs/math/0509308
http://front.math.ucdavis.edu/math.PR/0509308
(alternate)
Author(s): Ioannis Kontoyiannis (Athens U Econ & Business) and S.P. Meyn (Univ Ill. Urbana-Champaign)
Abstract: We continue the investigation of the spectral theory and exponential
asymptotics of Markov processes, following Kontoyiannis and Meyn (2003). We
introduce a new family of nonlinear Lyapunov drift criteria, characterizing
distinct subclasses of geometrically ergodic Markov processes in terms of
inequalities for the nonlinear generator. We concentrate on the class of
"multiplicatively regular" Markov processes, characterized via conditions
similar to (but weaker than) those of Donsker-Varadhan. For any such process
{Phi(t)} with transition kernel P on a general state space, the following are
obtained. 1. SPECTRAL THEORY: For a large class of functionals F, the kernel
Phat(x,dy) = e^{F(x)}P(x,dy) has a discrete spectrum in an appropriately
defined Banach space. There exists a "maximal" solution to the "multiplicative
Poisson equation," defined as the eigenvalue problem for Phat. Regularity
properties are established for \Lambda(F) = \log(\lambda), where \lambda is the
maximal eigenvalue, and for its convex dual. 2. MULTIPLICATIVE MEAN ERGODIC
THEOREM: The normalized mean E_x[\exp(S_t)] of the exponential of the partial
sums {S_t} of the process with respect to any one of the above functionals F,
converges to the maximal eigenfunction. 3. MULTIPLICATIVE REGULARITY: The drift
criterion under which our results are derived is equivalent to the existence of
regeneration times with finite exponential moments for {S_t}. 4. LARGE
DEVIATIONS: The sequence of empirical measures of {Phi(t)} satisfies an LDP in
a topology finer than the \tau-topology. The rate function is \Lambda^* and it
coincides with the Donsker-Varadhan rate function. 5. EXACTR LARGE DEVIATIONS:
The partial sums {S_t} satisfy an exact LD expansion, analogous to that
obtained for independent random variables.
http://arXiv.org/abs/math/0509310
http://front.math.ucdavis.edu/math.PR/0509310
(alternate)
Author(s): Alain Rouault
Abstract: This paper concentrates on asymptotic properties of determinants of some
random symmetric matrices. If B_{n,r} is a n x r rectangular matrix and
B_{n,r}' its transpose, we study det (B_{n,r}'B_{n,r}) when n,r tends to
infinity with r/n \to c\in (0,1). The r column vectors of B_{n,r} are chosen
independently, with common distribution \nu_n. The Wishart ensemble corresponds
to \nu_n = {\cal N}(0, I_n), the standard normal distribution. We call uniform
Gram ensemble the ensemble corresponding to \nu_n = \sigma_n, the uniform
distribution on the unit sphere `S_{n-1}. In the Wishart ensemble, a well known
Bartlett's theorem decomposes the above determinant into a product of
chi-square variables. The same holds in the uniform Gram ensemble. This allows
us to study the process \{\frac{1}{n}\log \det\big(B_{n,\lfloor
nt\rfloor}'B_{n,\lfloor nt\rfloor}\big), t \in [0,1]\} and its asymptotic
behavior as n\to \infty: a.s. convergence, fluctuations, large deviations. We
connect the results for marginals (fixed t) with those obtained by the spectral
method.
http://arXiv.org/abs/math/0509021
http://front.math.ucdavis.edu/math.PR/0509021
(alternate)
Author(s): X. Bressaud and A. Maass and S. Martinez and J. San Martin
Abstract: We study the standard property of the natural filtration associated to a 0-1
valued stationary process. In our main result we show that if the process has
summable memory decay, then the associated filtration is standard. We prove it
by coupling techniques. For a process whose associated filtration is standard
we construct a product type filtration extending it, based upon the usual
couplings and the Vershik's criterion for standardness.
http://arXiv.org/abs/math/0509317
http://front.math.ucdavis.edu/math.PR/0509317
(alternate)
Author(s): Mireille Bousquet-M\'{e}lou (LaBRI) and Svante Janson
Abstract: It has been known for a few years that the occupation measure of several
models of embedded trees converges, after a suitable normalization, to the
random measure called ISE (Integrated SuperBrownian Excursion). Here, we prove
a local version of this result: ISE has a (random) H\"{o}lder continuous
density, and the vertical profile of embedded trees converges to this density,
at least for some such trees. As a consequence, we derive a formula for the
distribution of the density of ISE at a given point. This follows from earlier
results by Bousquet-M\'{e}lou on convergence of the vertical profile at a fixed
point. We also provide a recurrence relation defining the moments of the
(random) moments of ISE.
http://arXiv.org/abs/math/0509322
http://front.math.ucdavis.edu/math.PR/0509322
(alternate)
Author(s): Jan van Neerven and Enrico Priola
Abstract: Let $E$ be a real Banach space. We study the Ornstein-Uhlenbeck semigroup
$P(t)$ associated with the Ornstein-Uhlenbeck operator $$ Lf(x) = \frac12 {\rm
Tr} Q D^2 f(x) + .$$ Here $Q$ is a positive symmetric operator from
$E^*$ to $E$ and $A$ is the generator of a $C_0$-semigroup $S(t)$ on $E$. Under
the assumption that $P$ admits an invariant measure $\mu$ we prove that if $S$
is eventually compact and the spectrum of its generator is nonempty, then $$\n
P(t)-P(s)\n_{L^1(E,\mu)} = 2$$ for all $t,s\ge 0$ with $t\not=s$. This result
is new even when $E = \R^n$. We also study the behaviour of $P$ in the space
$BUC(E)$. We show that if $A\not=0$ there exists $t_0>0$ such that $$\n
P(t)-P(s)\n_{BUC(E)} = 2$$ for all $0\le t,s\le t_0$ with $t\not=s$. Moreover,
under a nondegeneracy assumption or a strong Feller assumption, the following
dichotomy holds: either $$ \n P(t)- P(s)\n_{BUC(E)} = 2$$ for all $t,s\ge 0$, \
$t\not=s$, or $S$ is the direct sum of a nilpotent semigroup and a
finite-dimensional periodic semigroup. Finally we investigate the spectrum of
$L$ in the spaces $L^1(E,\mu)$ and $BUC(E)$.
http://arXiv.org/abs/math/0509309
http://front.math.ucdavis.edu/math.FA/0509309
(alternate)
Author(s): Annie Millet (PMA) and Marta Sanz-Sol\'{e}
Abstract: We consider a geometric rough path associated with a fractional Brownian
motion with Hurst parameter $H\in]{1/4}, {1/2}[$. We give an approximation
result in a modulus type distance, up to the second order, by means of a
sequence of rough paths lying above elements of the reproducing kernel Hilbert
space.
http://arXiv.org/abs/math/0509353
http://front.math.ucdavis.edu/math.PR/0509353
(alternate)
Author(s): Akimichi Takemura and Taiji Suzuki
Abstract: In this expository paper we illustrate the generality of game theoretic
probability protocols of Shafer and Vovk (2001) in finite-horizon discrete
games. By restricting ourselves to finite-horizon discrete games, we can
explicitly describe how discrete distributions with finite support and the
discrete pricing formulas, such as the Cox-Ross-Rubinstein formula, are
naturally derived from game-theoretic probability protocols. Corresponding to
any discrete distribution with finite support, we construct a finite-horizon
discrete game, a replicating strategy of Skeptic, and a neutral forecasting
strategy of Forecaster, such that the discrete distribution is derived from the
game. Construction of a replicating strategy is the same as in the standard
arbitrage arguments of pricing European options in the binomial tree models.
However the game theoretic framework is advantageous because no a priori
probabilistic assumption is needed.
http://arXiv.org/abs/math/0509367
http://front.math.ucdavis.edu/math.PR/0509367
(alternate)
Author(s): Xiaowen Zhou
Abstract: In this paper we consider a stepping-stone model on a circle with circular
Brownian migration. We first point out a connection between Arratia flow and
the marginal distribution of this model. We then give a new representation for
the stepping-stone model using Arratia flow and circular coalescing Brownian
motion. Such a representation enables us to carry out some explicit
computation. In particular, we find the Laplace transform for the time when
there is only a single type left across the circle.
http://arXiv.org/abs/math/0509383
http://front.math.ucdavis.edu/math.PR/0509383
(alternate)
Author(s): Itai Benjamini and Simi Haber and Michael Krivelevich and Eyal Lubetzky
Abstract: The isoperimetric constant of a graph $G$ on $n$ vertices, $i(G)$, is the
minimum of $\frac{|\partial S|}{|S|}$, taken over all nonempty subsets
$S\subset V(G)$ of size at most $n/2$, where $\partial S$ denotes the set of
edges with precisely one end in $S$. A random graph process on $n$ vertices,
$\widetilde{G}(t)$, is a sequence of $\binom{n}{2}$ graphs, where
$\widetilde{G}(0)$ is the edgeless graph on $n$ vertices, and
$\widetilde{G}(t)$ is the result of adding an edge to $\widetilde{G}(t-1)$,
uniformly distributed over all the missing edges. We show that in almost every
graph process $i(\widetilde{G}(t))$ equals the minimal degree of
$\widetilde{G}(t)$ as long as the minimal degree is $o(\log n)$. Furthermore,
we show that this result is essentially best possible, by demonstrating that
along the period in which the minimum degree is typically $\Theta(\log n)$, the
ratio between the isoperimetric constant and the minimum degree falls from 1 to
1/2, its final value.
http://arXiv.org/abs/math/0509022
http://front.math.ucdavis.edu/math.PR/0509022
(alternate)
Author(s): A. Bovier and A. Faggionato
Abstract: Sinai's walk can be thought of as a random walk on the set of interger
numbers with random potential V, with V weakly converging under diffusive
rescaling to a two-sided Brownian motion. We consider here the generator L_N of
Sinai's walk on [-N,N] with Dirichlet conditions on -N,N. By means of potential
theory, for each h>0 we show the relation between the spectral properties of
L_N for eigenvalues of order o(exp{-h N^{1/2}}) and the distribution of the
h-extrema of the rescaled potential V_N(x)=V(Nx)/N^{1/2} defined on [-1,1].
Information about the h-extrema of V_N is derived from a result of Neveu and
Pitman concerning the statistics of h-extrema of Brownian motion. As first
application of our results, we give a proof of a refined version of Sinai's
localization theorem.
http://arXiv.org/abs/math/0509385
http://front.math.ucdavis.edu/math.PR/0509385
(alternate)
Author(s): Yuri Kozitsky and Tatiana Pasurek
Abstract: A lattice system of interacting temperature loops, which is used in the
Euclidean approach to describe equilibrium thermodynamic properties of an
infinite system of interacting quantum particles performing anharmonic
oscillations (quantum anharmonic crystal), is considered. For this system, it
is proven that: (a) the set of tempered Gibbs measures is non-void and weakly
compact; (b) every Gibbs measure obeys an exponential integrability estimate,
the same for all such measures; (c) every Gibbs measure has a Lebowitz-Presutti
type support; (d) the set of all Gibbs measures is a singleton at high
temperatures. In the case of attractive interaction and one-dimensional
oscillations we prove that at low temperatures the system undergoes a phase
transition. The uniqueness of Gibbs measures due to strong quantum effects
(strong diffusivity) and at a nonzero external field are also proven in this
case. Thereby, a complete description of the properties of the set of all Gibbs
measures has been done, which essentially extends and refines the results
obtained so far for models of this type.
http://arXiv.org/abs/math-ph/0509036
http://front.math.ucdavis.edu/math-ph/0509036
(alternate)
Author(s): J. Englander and P. L. Simon
Abstract: Consider the KPP-type equation of the form $\Delta u+f(u)=0$, where $f:[0,1]
\to \mathbb R_{+}$ is a concave function. We prove for arbitrary dimensions
that there is no solution bounded in $(0,1)$. The significance of this result
from the point of view of probability theory is also discussed.
http://arXiv.org/abs/math/0509384
http://front.math.ucdavis.edu/math.AP/0509384
(alternate)
Author(s): Anna Celaya and Anant P. Godbole and Mandy Rae Schleifer
Abstract: The classical Erd\H os-Ko-Rado (EKR) Theorem states that if we choose a
family of subsets, each of size (k), from a fixed set of size (n (n > 2k)),
then the largest possible pairwise intersecting family has size (t ={n-1\choose
k-1}). We consider the probability that a randomly selected family of size
(t=t_n) has the EKR property (pairwise nonempty intersection) as $n$ and
$k=k_n$ tend to infinity, the latter at a specific rate. As $t$ gets large, the
EKR property is less likely to occur, while as $t$ gets smaller, the EKR
property is satisfied with high probability. We derive the threshold value for
$t$ using Janson's inequality. Using the Stein-Chen method we show that the
distribution of $X_0$, defined as the number of disjoint pairs of subsets in
our family, can be approximated by a Poisson distribution. We extend our
results to yield similar conclusions for $X_i$, the number of pairs of subsets
that overlap in exactly $i$ elements. Finally, we show that the joint
distribution $(X_0, X_1, ..., X_b)$ can be approximated by a multidimensional
Poisson vector with independent components.
http://arXiv.org/abs/math/0509382
http://front.math.ucdavis.edu/math.CO/0509382
(alternate)
Author(s): Chunrong Feng and Huaizhong Zhao
Abstract: In this paper, we prove two main results. The first one is to give a new
condition for the existence of two-parameter $p,q$-variation path integrals and
dominated convergence results for both the one-parameter and two-parameter
integrals. Our condition of locally bounded $p,q$-variation is more natural and
easy to verify than those of Young. The second result is to define the integral
of local time pathwise and then give generalized Ito's formula when
$\nabla^-f(s,x)$ is only of bounded $p,q$-variation in $(s,x)$. In the case
that $g(s,x)=\nabla^-f(s,x)$ is of locally bounded variation in $(s,x)$, the
integral $\int_{-\infty}^\infty\int_0^t \nabla^-f(s,x)d_{s,x}L_s(x)$ is the
Lebesgue-Stieltjes integral and was used in Elworthy, Truman and Zhao (2004).
When $g(s,x)=\nabla^-f(s,x)$ is of only locally $p, q$-variation, where $p\geq
1$,$q\geq 1$, and $2q+1>2pq$, the integral is a two-parameter $p,1$-variation
path integral rather than a Lebesgue-Stieltjes integral. In the special case
that $f(s,x)=f(x)$ is independent of $s$, we give a new condition for Meyer's
formula and $\int_{-\infty}^\infty L_t(x)d_x\nabla^-f(x)$ is defined pathwise
as a Lyons-Young's integral of $p$-variation. For this we prove the local time
$L_t(x)$ is of $p$-variation in $x$ for each $t\geq 0$, for each $p>2$ almost
surely ($p$-variation in the sense of Young. Both results are new in rough path
theory and local time integration respectively.
http://arXiv.org/abs/math/0509422
http://front.math.ucdavis.edu/math.PR/0509422
(alternate)
Author(s): D. Marinucci
Abstract: The angular bispectrum of spherical random fields has recently gained an
enormous importance, especially in connection with statistical inference on
cosmological data. In this paper, we provide expressions for its moments of
arbitrary order and we use these results to establish a multivariate central
limit theorem and higher order approximations. The results rely upon
combinatorial methods from graph theory and a detailed investigation for the
asymptotic behaviour of Clebsch-Gordan coefficients; the latter are widely used
in representation theory and quantum theory of angular momentum.
http://arXiv.org/abs/math/0509430
http://front.math.ucdavis.edu/math.PR/0509430
(alternate)
Author(s): Diethelm Wuertz and Helmut G. Katzgraber
Abstract: It is well known that the finite-sample null distribution of the Jarque-Bera
Lagrange Multiplier (LM) test for normality and its adjusted version (ALM)
introduced by Urzua differ considerably from their asymptotic chi^2(2) limit.
Here, we present results from Monte Carlo simulations using 10^7 replications
which yield very precise numbers for the LM and ALM statistic over a wide range
of critical values and sample sizes. This enables a precise implementation of
the Jarque-Bera LM and ALM test for finite samples.
http://arXiv.org/abs/math/0509423
http://front.math.ucdavis.edu/math.ST/0509423
(alternate)
Author(s): Elizabeth Meckes
Abstract: Let $M$ be a random matrix in the orthogonal group $\O_n$, distributed
according to Haar measure, and let $A$ be a fixed $n\times n$ matrix over $\R$
such that $\tr(AA^t)=n$. Then the total variation distance of the random
variable $\tr(AM)$ to standard normal is bounded by $2\sqrt{3}/(n-1)$, and this
rate is sharp up to the constant. Analogous results are obtained for $M$ a
random unitary matrix and $A$ a fixed $n\times n$ matrix over $\C$. The proofs
are via an improvement of Stein's method of exchangeable pairs which makes use
of the continuous nature of the symmetries of the classical matrix groups.
http://arXiv.org/abs/math/0509441
http://front.math.ucdavis.edu/math.PR/0509441
(alternate)
Author(s): Larry Goldstein and Aihua Xia
Abstract: We introduce a new family of distributions to approximate $\prob(W\in A)$ for
$A\subset\{...,-2,-1,0,1,2,...\}$ and $W$ a sum of independent integer-valued
random variables $\xi_1$, $\xi_2$, $...$, $\xi_n$ with finite second moments,
where with large probability $W$ is not concentrated on a lattice of span
greater than 1. The well-known Berry--Esseen theorem states that for $Z$ a
normal random variable with mean $\mean(W)$ and variance $\var(W)$, $\prob(Z
\in A)$ provides a good approximation to $\prob(W \in A)$ for $A$ of the form
$(-\infty,x]$. However, for more general $A$ such as the set of all even
numbers, the normal approximation becomes unsatisfactory and it is desirable to
have an appropriate discrete, non-normal, distribution which approximates $W$
in total variation, and a discrete version of the Berry--Esseen theorem to
bound the error. In this paper, using the concept of zero biasing for discrete
random variables [cf Goldstein and Reinert (2005)], we introduce a new family
of discrete distributions and provide a discrete version of the Berry--Esseen
theorem showing how members of the family approximate the distribution of a sum
$W$ of integer valued variables in total variation.
http://arXiv.org/abs/math/0509444
http://front.math.ucdavis.edu/math.PR/0509444
(alternate)
Author(s): Luigi Manca
Abstract: We consider stochastic semilinear partial differential equations with
Lipschitz nonlinear terms. We prove existence and uniqueness of an invariant
measure and the existence of a solution for the corresponding Kolmogorov
equation in the space $L^2(H;\nu)$, where $\nu$ is the invariant measure. We
also prove the closability of the derivative operator and an integration by
parts formula. Finally, under boundness conditions on the nonlinear term, we
prove a Poincar\'e inequality, a logarithmic Sobolev inequality and the
ipercontractivity of the transition semigroup.
http://arXiv.org/abs/math/0509446
http://front.math.ucdavis.edu/math.PR/0509446
(alternate)
Author(s): D. Marinucci
Abstract: The angular bispectrum of spherical random fields has recently gained an
enormous importance, especially in connection with statistical inference on
cosmological data. In this paper, we provide expressions for its moments of
arbitrary order and we use these results to establish a multivariate central
limit theorem and higher order approximations. The results rely upon
combinatorial methods from graph theory and a detailed investigation for the
asymptotic behaviour of Clebsch-Gordan coefficients; the latter are widely used
in representation theory and quantum theory of angular momentum.
http://arXiv.org/abs/math/0509430
http://front.math.ucdavis.edu/math.PR/0509430
(alternate)
Author(s): Itai Benjamini and Krzysztof Burdzy and Zhen-Qing Chen
Abstract: A pair of Markov processes is called a Markov coupling if both processes have
the same transition probabilities and the pair is also a Markov process. We say
that a coupling is ``shy'' if the processes never come closer than some
(random) strictly positive distance from each other. We investigate whether shy
couplings exist for several classes of Markov processes.
http://arXiv.org/abs/math/0509458
http://front.math.ucdavis.edu/math.PR/0509458
(alternate)
Author(s): Gideon Amir and Itai Benjamini and Gady Kozma
Abstract: We analyze random walk in the upper half of a three dimensional lattice which
goes down whenever it encounters a new vertex, a.k.a. excited random walk. We
show that it is recurrent with an expected number of returns of square-root log
n.
http://arXiv.org/abs/math/0509464
http://front.math.ucdavis.edu/math.PR/0509464
(alternate)
Author(s): Svante Janson
Abstract: Consider a random recusive tree with n vertices. We show that the number of
vertices with even depth is asymptotically normal as n tends to infinty. The
same is true for the number of vertices of depth divisible by m for m=3, 4 or
5; in all four cases the variance grows linearly. On the other hand, for m at
least 7, the number is not asymptotically normal, and the variance grows faster
than linear in n. The case m=6 is intermediate: the number is asymptotically
normal but the variance is of order n log n.
This is a simple and striking example of a type of phase transition that has
been observed by other authors in several cases. We prove, and perhaps explain,
this non-intuitive behavious using a translation to a generalized Polya urn.
Similar results hold for a random binary search tree; now the number of
vertices of depth divisible by m is asymptotically normal for m at most 8 but
not for m at least 9, and the variance grows linearly in the first case both
faster in the second. (There is no intermediate case.)
In contrast, we show that for conditioned Galton-Watson trees, including
random labelled trees and random binary trees, there is no such phase
transition: the number is asymptotically normal for every m.
http://arXiv.org/abs/math/0509471
http://front.math.ucdavis.edu/math.PR/0509471
(alternate)
Author(s): David Aldous and Maxim Krikun
Abstract: We prove consistency of four different approaches to formalizing the idea of
minimum average edge-length in a path linking some infinite subset of points of
a Poisson process. The approaches are (i) shortest path from origin through
some $m$ distinct points; (ii) shortest average edge-length in paths across the
diagonal of a large cube; (iii) shortest path through some specified proportion
$\delta$ of points in a large cube; (iv) translation-invariant measures on
paths in $\Reals^d$ which contain a proportion $\delta$ of the Poisson points.
We develop basic properties of a normalized average length function $c(\delta)$
and pose challenging open problem
http://arXiv.org/abs/math/0509492
http://front.math.ucdavis.edu/math.PR/0509492
(alternate)
Author(s): Jaksa Cvitanic and Robert Liptser and Boris Rozovskii
Abstract: This paper is concerned with nonlinear filtering of the coefficients in asset
price models with stochastic volatility. More specifically, we assume that the
asset price process $ S=(S_{t})_{t\geq0} $ is given by \[
dS_{t}=r(\theta_{t})S_{t}dt+v(\theta_{t})S_{t}dB_{t}, \] where
$B=(B_{t})_{t\geq0}$ is a Brownian motion, $v$ is a positive function, and
$\theta=(\theta_{t})_{t\geq0}$ is a c\'{a}dl\'{a}g strong Markov process. The
random process $\theta$ is unobservable. We assume also that the asset price
$S_{t}$ is observed only at random times $0<\tau_{1}<\tau_{2}<....$ This is an
appropriate assumption when modelling high frequency financial data (e.g.,
tick-by-tick stock prices).
In the above setting the problem of estimation of $\theta$ can be approached
as a special nonlinear filtering problem with measurements generated by a
multivariate point process $(\tau_{k},\log S_{\tau_{k}})$. While quite natural,
this problem does not fit into the standard diffusion or simple point process
filtering frameworks and requires more technical tools. We derive a closed form
optimal recursive Bayesian filter for $\theta_{t}$, based on the observations
of $(\tau_{k},\log S_{\tau_{k}})_{k\geq1}$. It turns out that the filter is
given by a recursive system that involves only deterministic Kolmogorov-type
equations, which should make the numerical implementation relatively easy.
http://arXiv.org/abs/math/0509503
http://front.math.ucdavis.edu/math.PR/0509503
(alternate)
Author(s): Fabrice Baudoin and Laure Coutin
Abstract: In this paper, by using a Taylor development type formula, we show how it is
possible to associate differential operators with stochastic differential
equations driven by a fractional Brownian motion. As an application, we deduce
that invariant measures for such SDEs must satisfy an infinite dimensional
system of partial differential equations.
http://arXiv.org/abs/math/0509511
http://front.math.ucdavis.edu/math.PR/0509511
(alternate)
Author(s): Thomas Duquesne (Paris 11) and Matthias Winkel (Oxford)
Abstract: We construct random locally compact real trees called Levy trees that are the
genealogical trees associated with continuous-state branching processes. More
precisely, we define a growing family of discrete Galton-Watson trees with
i.i.d. exponential branch lengths that is consistent under Bernoulli
percolation on leaves; we define the Levy tree as the limit of this growing
family with respect to the Gromov-Hausdorff topology on metric spaces. This
elementary approach notably includes supercritical trees and does not make use
of the height process introduced by Le Gall and Le Jan to code the genealogy of
(sub)critical continuous-state branching processes. We construct the mass
measure of Levy trees and we give a decomposition along the ancestral subtree
of a Poisson sampling directed by the mass measure.
http://arXiv.org/abs/math/0509518
http://front.math.ucdavis.edu/math.PR/0509518
(alternate)
Author(s): Thomas Duquesne (Paris11)
Abstract: We study a genealogical model for continuous-state branching processes with
immigration with a (sub)critical branching mechanism. This model allows the
immigrants to be on the same line of descent. The corresponding family tree is
an ordered rooted continuum random tree with a single infinite end defined
thanks to two continuous processes denoted by $(\overleftarrow{H}_t ;t\geq 0)$
and $(\overrightarrow{H}_t ;t\geq 0)$ that code the parts at resp. the left and
the right hand of the infinite line of descent of the tree. These processes are
called the left and the right height processes. We define their local time
processes via an approximation procedure and we prove that they enjoy a
Ray-Knight property. We also discuss the important special case corresponding
to the size-biased Galton-Watson tree in the continuous setting. In the last
part of the paper we give a convergence result under general assumptions for
rescaled discrete left and right contour processes of sequences of
Galton-Watson trees with immigration. We also provide a strong invariance
principle for a sequence of rescaled Galton-Watson processes with immigration
that also holds in the supercritical case.
http://arXiv.org/abs/math/0509519
http://front.math.ucdavis.edu/math.PR/0509519
(alternate)
Author(s): Thomas Duquesne
Abstract: Let $X$ be a real L\'evy process and let $\Xpos $ be the process conditioned
to stay positive. We assume that $ 0 $ is regular for $(-\infty, 0)$ and $(0,
+\infty) $ with respect to $X$. Using elementary excursion theory arguments, we
provide a simple probabilistic description of the reversed paths of $X$ and
$\Xpos $ at their first hitting time of $ (x, +\infty)$ and last passage time
of $ (-\infty, x ] $, on a fixed time interval $[0, t]$, for a positive level
$x$. From these reversion formulas, we derive an extension to general L\'evy
processes of Williams' decomposition theorems, Bismut's decomposition of the
excursion above the infimum and also several relations involving the reversed
excursion under the maximum.
http://arXiv.org/abs/math/0509520
http://front.math.ucdavis.edu/math.PR/0509520
(alternate)
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