Showing papers in "Probability Theory and Related Fields in 2005"

Strong solutions of stochastic equations with singular time dependent drift

Abstract: We prove existence and uniqueness of strong solutions to stochastic equations in domains with unit diffusion and singular time dependent drift b up to an explosion time. We only assume local L q _L p -integrability of b in ℝ×G with d/p+2/q<1. We also prove strong Feller properties in this case. If b is the gradient in x of a nonnegative function ψ blowing up as G∋x→∂G, we prove that the conditions 2D t ψ≤Kψ,2D t ψ+Δψ≤Ke ɛψ ,ɛ ∈ [0,2), imply that the explosion time is infinite and the distributions of the solution have sub Gaussian tails.

Probabilistic and fractal aspects of Lévy trees

Abstract: We investigate the random continuous trees called Levy trees, which are obtained as scaling limits of discrete Galton-Watson trees. We give a mathematically precise definition of these random trees as random variables taking values in the set of equivalence classes of compact rooted ℝ-trees, which is equipped with the Gromov-Hausdorff distance. To construct Levy trees, we make use of the coding by the height process which was studied in detail in previous work. We then investigate various probabilistic properties of Levy trees. In particular we establish a branching property analogous to the well-known property for Galton-Watson trees: Conditionally given the tree below level a, the subtrees originating from that level are distributed as the atoms of a Poisson point measure whose intensity involves a local time measure supported on the vertices at distance a from the root. We study regularity properties of local times in the space variable, and prove that the support of local time is the full level set, except for certain exceptional values of a corresponding to local extinctions. We also compute several fractal dimensions of Levy trees, including Hausdorff and packing dimensions, in terms of lower and upper indices for the branching mechanism function ψ which characterizes the distribution of the tree. We finally discuss some applications to super-Brownian motion with a general branching mechanism.

New dependence coefficients. Examples and applications to statistics

Abstract: To measure the dependence between a real-valued random variable X and a σ-algebra , we consider four distances between the conditional distribution function of X given and the distribution function of X. The coefficients obtained are weaker than the corresponding mixing coefficients and may be computed in many situations. In particular, we show that they are well adapted to functions of mixing sequences, iterated random functions and dynamical systems. Starting from a new covariance inequality, we study the mean integrated square error for estimating the unknown marginal density of a stationary sequence. We obtain optimal rates for kernel estimators as well as projection estimators on a well localized basis, under a minimal condition on the coefficients. Using recent results, we show that our coefficients may be also used to obtain various exponential inequalities, a concentration inequality for Lipschitz functions, and a Berry-Esseen type inequality.

Glauber dynamics on trees and hyperbolic graphs

Abstract: We study continuous time Glauber dynamics for random configurations with local constraints (eg proper coloring, Ising and Potts models) on finite graphs with n vertices and of bounded degree We show that the relaxation time (defined as the reciprocal of the spectral gap |λ1−λ2|) for the dynamics on trees and on planar hyperbolic graphs, is polynomial in n For these hyperbolic graphs, this yields a general polynomial sampling algorithm for random configurations We then show that for general graphs, if the relaxation time τ2 satisfies τ2=O(1), then the correlation coefficient, and the mutual information, between any local function (which depends only on the configuration in a fixed window) and the boundary conditions, decays exponentially in the distance between the window and the boundary For the Ising model on a regular tree, this condition is sharp

Evolving sets, mixing and heat kernel bounds

Abstract: We show that a new probabilistic technique, recently introduced by the first author, yields the sharpest bounds obtained to date on mixing times of Markov chains in terms of isoperimetric properties of the state space (also known as conductance bounds or Cheeger inequalities). We prove that the bounds for mixing time in total variation obtained by Lovasz and Kannan, can be refined to apply to the maximum relative deviation |p n (x,y)/π(y)−1| of the distribution at time n from the stationary distribution π. We then extend our results to Markov chains on infinite state spaces and to continuous-time chains. Our approach yields a direct link between isoperimetric inequalities and heat kernel bounds; previously, this link rested on analytic estimates known as Nash inequalities.

Discrete Quantum Walks Hit Exponentially Faster

Abstract: This paper addresses the question: what processes take polynomial time on a quantum computer that require exponential time classically? We show that the hitting time of the discrete time quantum walk on the n-bit hypercube from one corner to its opposite is polynomial in n. This gives the first exponential quantum-classical gap in the hitting time of discrete quantum walks. We provide the basic framework for quantum hitting time and give two alternative definitions to set the ground for its study on general graphs. We outline a possible application to sequential packet routing.

Product of random projections, Jacobi ensembles and universality problems arising from free probability

Abstract: We consider the product of two independent randomly rotated projectors. The square of its radial part turns out to be distributed as a Jacobi ensemble. We study its global and local properties in the large dimension scaling relevant to free probability theory. We establish asymptotics for one point and two point correlation functions, as well as properties of largest and smallest eigenvalues.

A study of a class of stochastic differential equations with non-Lipschitzian coefficients

Abstract: We study a class of stochastic differential equations with non-Lipschitz coefficients. A unique strong solution is obtained and the non confluence of the solutions of stochastic differential equations is proved. The dependence with respect to the initial values is investigated. To obtain a continuous version of solutions, the modulus of continuity of coefficients is assumed to be less than |x-y| log MediaObjects/s00440-004-0398-zflb1.gif Finally a large deviation principle of Freidlin-Wentzell type is also established in the paper.

Interpretations of some parameter dependent generalizations of classical matrix ensembles

Abstract: Two types of parameter dependent generalizations of classical matrix ensembles are defined by their probability density functions (PDFs). As the parameter is varied, one interpolates between the eigenvalue PDF for the superposition of two classical ensembles with orthogonal symmetry and the eigenvalue PDF for a single classical ensemble with unitary symmetry, while the other interpolates between a classical ensemble with orthogonal symmetry and a classical ensemble with symplectic symmetry. We give interpretations of these PDFs in terms of probabilities associated to the continuous Robinson-Schensted-Knuth correspondence between matrices, with entries chosen from certain exponential distributions, and non-intersecting lattice paths, and in the course of this probability measures on partitions and pairs of partitions are identified. The latter are generalized by using Macdonald polynomial theory, and a particular continuum limit – the Jacobi limit – of the resulting measures is shown to give PDFs related to those appearing in the work of Anderson on the Selberg integral, and also in some classical work of Dixon. By interpreting Anderson’s and Dixon’s work as giving the PDF for the zeros of a certain rational function, it is then possible to identify random matrices whose eigenvalue PDFs realize the original parameter dependent PDFs. This line of theory allows sampling of the original parameter dependent PDFs, their Dixon-Anderson-type generalizations and associated marginal distributions, from the zeros of certain polynomials defined in terms of random three term recurrences.

Stochastic nonlinear beam equations

Abstract: An extensible beam equation with a stochastic force of a white noise type is studied, Lyapunov functions techniques being used to prove existence of global mild solutions and asymptotic stability of the zero solution.

Central limit theorems for random polytopes

Abstract: Let K be a smooth convex set. The convex hull of independent random points in K is a random polytope. Central limit theorems for the volume and the number of i dimensional faces of random polytopes are proved as the number of random points tends to infinity. One essential step is to determine the precise asymptotic order of the occurring variances.

Self-similar fragmentations derived from the stable tree II: splitting at nodes

Abstract: We study a natural fragmentation process of the so-called stable tree introduced by Duquesne and Le Gall, which consists in removing the nodes of the tree according to a certain procedure that makes the fragmentation self-similar with positive index. Explicit formulas for the semigroup are given, and we provide asymptotic results. We also give an alternative construction of this fragmentation, using paths of Levy processes, hence echoing the two alternative constructions of the standard additive coalescent by fragmenting the Brownian continuum random tree or using Brownian paths, respectively due to Aldous-Pitman and Bertoin.

An almost sure invariance principle for random walks in a space-time random environment

Abstract: We consider a discrete time random walk in a space-time i.i.d. random environ- ment. We use a martingale approach to show that the walk is diffusive in almost every fixed environment. We improve on existing results by proving an invariance principle and con- sidering environments with an L 2 averaged drift. We also state an a.s. invariance principle for random walks in general random environments whose hypothesis requires a subdiffusive bound on the variance of the quenched mean, under an ergodic invariant measure for the environment chain.

On estimating the derivatives of symmetric diffusions in stationary random environment, with applications to ∇ϕ interface model

Abstract: We consider diffusions on ℝ d or random walks on ℤ d in a random environment which is stationary in space and in time and with symmetric and uniformly elliptic coefficients. We show existence and Holder continuity of second space derivatives and time derivatives for the annealed kernels of such diffusions and give estimates for these derivatives. In the case of random walks, these estimates are applied to the Ginzburg-Landau ∇ϕ interface model.

Multi-excited random walks on integers

Abstract: We introduce a class of nearest-neighbor integer random walks in random and non-random media, which includes excited random walks considered in the literature. At each site the random walker has a drift to the right, the strength of which depends on the environment at that site and on how often the walker has visited that site before. We give exact criteria for recurrence and transience and consider the speed of the walk.

BSDEs with two reflecting barriers: the general result

Abstract: In this paper we show the existence of a solution for the BSDE with two reflecting barriers when those latter are completely separated. Neither Mokobodzki’s condition nor the regularity of the barriers are supposed. The main tool is the notion of local solution of reflected BSDEs. Applications related to Dynkin games and double obstacle variational inequality are given.

Modified logarithmic Sobolev inequalities and transportation inequalities

Abstract: We present a new class of modified logarithmic Sobolev inequality, interpolating between Poincare and logarithmic Sobolev inequalities, suitable for measures of the type $\exp(-|x|^\al)$ or $\exp(-|x|^\al\log^\beta(2+|x|))$ ($\al\in]1,2[$ and $\be\in\dR$) which lead to new concentration inequalities. These modified inequalities share common properties with usual logarithmic Sobolev inequalities, as tensorisation or perturbation, and imply as well Poincare inequality. We also study the link between these new modified logarithmic Sobolev inequalities and transportation inequalities.

On singularity of energy measures on self-similar sets

Abstract: We provide general criteria for energy measures of regular Dirichlet forms on self-similar sets to be singular to Bernoulli type measures. In particular, every energy measure is proved to be singular to the Hausdorff measure for canonical Dirichlet forms on 2-dimensional Sierpinski carpets.

Limit theorems for sums of random exponentials

Abstract: We study limiting distributions of exponential sums Open image in new window as t→∞, N→∞, where (Xi) are i.i.d. random variables. Two cases are considered: (A) ess sup Xi = 0 and (B) ess sup Xi = ∞. We assume that the function h(x)= -log P{Xi>x} (case B) or h(x) = -log P {Xi>-1/x} (case A) is regularly varying at ∞ with index 1 < ϱ <∞ (case B) or 0 < ϱ < ∞ (case A). The appropriate growth scale of N relative to t is of the form Open image in new window, where the rate function H0(t) is a certain asymptotic version of the function Open image in new window (case B) or Open image in new window (case A). We have found two critical points, λ1<λ2, below which the Law of Large Numbers and the Central Limit Theorem, respectively, break down. For 0 < λ < λ2, under the slightly stronger condition of normalized regular variation of h we prove that the limit laws are stable, with characteristic exponent α = α (ϱ, λ) ∈ (0,2) and skewness parameter β ≡ 1.

Conditional moments of q-Meixner processes

Abstract: We show that stochastic processes with linear conditional expectations and quadratic conditional variances are Markov, and their transition probabilities are related to a three-parameter family of orthogonal polynomials which generalize the Meixner polynomials. Special cases of these processes are known to arise from the non-commutative generalizations of the Levy processes.

L p estimates for the uniform norm of solutions of quasilinear SPDE's

Abstract: In this paper we prove Lp estimates (p≥2) for the uniform norm of the paths of solutions of quasilinear stochastic partial differential equations (SPDE) of parabolic type. Our method is based on a version of Moser's iteration scheme developed by Aronson and Serrin in the context of non-linear parabolic PDE.

Scaling limits of equilibrium wetting models in (1+1)–dimension

Abstract: We study the path properties for the δ-pinning wetting model in (1+1)–dimension. In other terms, we consider a random walk model with fairly general continuous increments conditioned to stay in the upper half plane and with a δ-measure reward for touching zero, that is the boundary of the forbidden region. It is well known that such a model displays a localization/delocalization transition, according to the size of the reward. Our focus is on getting a precise pathwise description of the system, in both the delocalized phase, that includes the critical case, and in the localized one. From this we extract the (Brownian) scaling limits of the model.

Weighted BMO and discrete time hedging within the Black-Scholes model

Abstract: The paper combines two objects rather different at first glance: spaces of stochastic processes having weighted bounded mean oscillation (weighted BMO) and the approximation of certain stochastic integrals, driven by the geometric Brownian motion, by integrals over piece-wise constant integrands. The consideration of the approximation error with respect to weighted BMO implies Lp and uniform distributional estimates for the approximation error by a John-Nirenberg type theorem. The general results about weighted BMO are given in the first part of the paper and applied to our approximation problem in the second one.

A local limit theorem for random walks conditioned to stay positive

Abstract: We consider a real random walk S_n = X_1 + ... + X_n attracted (without centering) to the normal law: this means that for a suitable norming sequence a_n we have the weak convergence S_n / a_n --> f(x) dx, where f(x) is the standard normal density (this happens in particular by the CLT when X_1 has zero mean and finite variance \sigma^2, with a_n = \sigma \sqrt{n}). A local refinement of this convergence is provided by Gnedenko's and Stone's Local Limit Theorems, in the lattice and nonlattice case respectively. Now let C_n denote the event (S_1 > 0, ... , S_n > 0) and let S_n^+ denote the random variable S_n conditioned on C_n: it is known that S_n^+ / a_n --> f^+(x) \dd x, where f^+(x) := x \exp(-x^2/2) \ind_{x > 0}. What we establish in this paper is an equivalent of Gnedenko's and Stone's Local Limit Theorems for this weak convergence. We also consider the particular case when X_1 has an absolutely continuous law: in this case the uniform convergence of the density of S_n^+ / a_n towards f^+(x) holds under a standard additional hypothesis, in analogy to the classical case. We finally discuss an application of our main results to the asymptotic behavior of the joint renewal measure of the ladder variables process. Unlike the classical proofs of the LLT, we make no use of characteristic functions: our techniques are rather taken from the so--called Fluctuation Theory for random walks.

On the central limit theorem for geometrically ergodic Markov chains

Abstract: Let X-0,X-1,... be a geometrically ergodic Markov chain with state space X and stationary distribution pi. It is known that if h:X -> R satisfies pi(vertical bar h vertical bar(2+epsilon)) 0, then the normalized sums of the X-i's obey a central limit theorem. Here we show, by means of a counterexample, that the condition pi(vertical bar h vertical bar(2+epsilon)) < infinity cannot be weakened to only assuming a finite second moment, i.e., pi(h(2)) < infinity.

Estimates on path delocalization for copolymers at selective interfaces

Abstract: Starting from the simple symmetric random walk {Sn}n, we introduce a new process whose path measure is weighted by a factor exp Open image in new window with α,h≥0, {Wn}n a typical realization of an IID process and N a positive integer. We are looking for results in the large N limit. This factor favors Sn>0 if Wn+h>0 and Sn<0 if Wn+h<0. The process can be interpreted as a model for a random heterogeneous polymer in the proximity of an interface separating two selective solvents. It has been shown [6] that this model undergoes a (de)localization transition: more precisely there exists a continuous increasing function λ↦hc(λ) such that if h

Branching-coalescing particle systems

Abstract: We study the ergodic behavior of systems of particles performing independent random walks, binary splitting, coalescence and deaths. Such particle systems are dual to systems of linearly interacting Wright-Fisher diffusions, used to model a population with resampling, selection and mutations. We use this duality to prove that the upper invariant measure of the particle system is the only homogeneous nontrivial invariant law and the limit started from any homogeneous nontrivial initial law.

Maxima of entries of Haar distributed matrices

Abstract: Let Γ n =(γ ij ) be an n×n random matrix such that its distribution is the normalized Haar measure on the orthogonal group O(n). Let also W n :=max1≤ i , j ≤ n |γ ij |. We obtain the limiting distribution and a strong limit theorem on W n . A tool has been developed to prove these results. It says that up to n/( log n)2 columns of Γ n can be approximated simultaneously by those of some Y n =(y ij ) in which y ij are independent standard normals. Similar results are derived also for the unitary group U(n), the special orthogonal group SO(n), and the special unitary group SU(n).

Functional central limit theorems for a large network in which customers join the shortest of several queues

Abstract: We consider N single server infinite buffer queues with service rate β. Customers arrive at rate Nα, choose L queues uniformly, and join the shortest. We study the processes Open image in new window for large N, where RNt(k) is the fraction of queues of length at least k at time t. Laws of large numbers (LLNs) are known, see Vvedenskaya et al. [15], Mitzenmacher [12] and Graham [5]. We consider certain Hilbert spaces with the weak topology. First, we prove a functional central limit theorem (CLT) under the a priori assumption that the initial data RN0 satisfy the corresponding CLT. We use a compactness-uniqueness method, and the limit is characterized as an Ornstein-Uhlenbeck (OU) process. Then, we study the RN in equilibrium under the stability condition α<β, and prove a functional CLT with limit the OU process in equilibrium. We use ergodicity and justify the inversion of limits lim N→∞ lim t→∞= lim t→∞ lim N→∞ by a compactness-uniqueness method. We deduce a posteriori the CLT for RN0 under the invariant laws, an interesting result in its own right. The main tool for proving tightness of the implicitly defined invariant laws in the CLT scaling and ergodicity of the limit OU process is a global exponential stability result for the nonlinear dynamical system obtained in the functional LLN limit.

Distributions diophantiennes et théorème limite local sur

Abstract: We study the speed of convergence of nd/2∫fdμ*n in the local limit theorem on Open image in new window under very general conditions upon the function f and the distribution μ. We show that this speed is at least of order Open image in new window and we give a simple characterization (in diophantine terms) of those measures for which this speed (and the full local Edgeworth expansion) holds for smooth enough f. We then derive a uniform local limit theorem for moderate deviations under a mild moment assumption. This in turn yields other limit theorems when f is no longer assumed integrable but only bounded and Lipschitz or Holder. We finally give an application to equidistribution of random walks.