Showing papers in "Probability Theory and Related Fields in 2004"

Time-fractional telegraph equations and telegraph processes with brownian time

Abstract: We study the fundamental solutions to time-fractional telegraph equations of order 2α. We are able to obtain the Fourier transform of the solutions for any α and to give a representation of their inverse, in terms of stable densities. For the special case α=1/2, we can show that the fundamental solution is the distribution of a telegraph process with Brownian time. In a special case, this becomes the density of the iterated Brownian motion, which is therefore the fundamental solution to a fractional diffusion equation of order 1/2 with respect to time.

The Brownian loop soup

Abstract: We define a natural conformally invariant measure on unrooted Brownian loops in the plane and study some of its properties. We relate this measure to a measure on loops rooted at a boundary point of a domain and show how this relation gives a way to ‘‘chronologically add Brownian loops’’ to simple curves in the plane.

Quenched invariance principles for walks on clusters of percolation or among random conductances

Abstract: In this work we principally study random walk on the supercritical infinite cluster for bond percolation on ℤd. We prove a quenched functional central limit theorem for the walk when d≥4. We also prove a similar result for random walk among i.i.d. random conductances along nearest neighbor edges of ℤd, when d≥1.

Random planar lattices and integrated superBrownian excursion

Abstract: In this extended abstract, a surprising connection is described between a specific brand of random lattices, namely planar quadrangulations, and Aldous' Integrated SuperBrownian Excursion (ISE). As a consequence, the radius rn of a random quadrangulation with n faces is shown to converge, up to scaling, to the width r = R - L of the support of the one-dimensional ISE, or precisely: $$ n^{ - {1 \mathord{\left/ {\vphantom {1 4}} \right. \kern- ulldelimiterspace} 4}} r_n \xrightarrow{{law}} (8/9)^{{1 \mathord{\left/ {\vphantom {1 4}} \right. \kern- ulldelimiterspace} 4}} r. $$

Fisher Information inequalities and the Central Limit Theorem

Abstract: We give conditions for an O(1/n) rate of convergence of Fisher information and relative entropy in the Central Limit Theorem. We use the theory of projections in L 2 spaces and Poincare inequalities, to provide a better understanding of the decrease in Fisher information implied by results of Barron and Brown. We show that if the standardized Fisher information ever becomes finite then it converges to zero.

Monge’s problem with a quadratic cost by the zero-noise limit of h-path processes

Abstract: We study the asymptotic behavior, in the zero-noise limit, of solutions to Schrodinger's functional equations and that of h-path processes, and give a new proof of the existence of the minimizer of Monge's problem with a quadratic cost.

Central limit theorem and stable laws for intermittent maps

Abstract: In the setting of abstract Markov maps, we prove results concerning the convergence of renormalized Birkhoff sums to normal laws or stable laws. They apply to one-dimensional maps with a neutral fixed point at 0 of the form x+x 1+α , for α∈(0, 1). In particular, for α>1/2, we show that the Birkhoff sums of a Holder observable f converge to a normal law or a stable law, depending on whether f(0)=0 or f(0)≠0. The proof uses spectral techniques introduced by Sarig, and Wiener’s Lemma in non-commutative Banach algebras.

Monge-Kantorovitch Measure Transportation and Monge-Ampère Equation on Wiener Space

Abstract: Let (W,μ,H) be an abstract Wiener space assume two ν i ,i=1,2 probabilities on (W,ℬ(W)). We give some conditions for the Wasserstein distance between ν1 and ν2 with respect to the Cameron-Martin space to be finite, where the infimum is taken on the set of probability measures β on W×W whose first and second marginals are ν1 and ν2. In this case we prove the existence of a unique (cyclically monotone) map T=I W +ξ, with ξ:W→H, such that T maps ν1 to ν2. Moreover, if ν2≪μ, then T is stochastically invertible, i.e., there exists S:W→W such that S○T=I W ν1 a.s. and T○S=I W ν2 a.s. If, in addition, ν1=μ, then there exists a 1-convex function φ in the Gaussian Sobolev space such that ξ=∇φ. These results imply that the quasi-invariant transformations of the Wiener space with finite Wasserstein distance from μ can be written as the composition of a transport map T and a rotation, i.e., a measure preserving map. We give also 1-convex sub-solutions and Ito-type solutions of the Monge-Ampere equation on W.

A series expansion of fractional Brownian motion

Abstract: Let B be a fractional Brownian motion with Hurst index H∈(0,1). Denote by the positive, real zeros of the Bessel function J −H of the first kind of order −H, and let be the positive zeros of J 1−H . In this paper we prove the series representation where X 1 ,X 2 ,... and Y 1 ,Y 2 ,... are independent, Gaussian random variables with mean zero and and the constant c H 2 is defined by c H 2 =π−1Γ(1+2H) sin πH. We show that with probability 1, both random series converge absolutely and uniformly in t∈[0,1], and we investigate the rate of convergence.

A proof of Parisi’s conjecture on the random assignment problem

Abstract: An assignment problem is the optimization problem of finding, in an m by n matrix of nonnegative real numbers, k entries, no two in the same row or column, such that their sum is minimal. Such an optimization problem is called a random assignment problem if the matrix entries are random variables. We give a formula for the expected value of the optimal k-assignment in a matrix where some of the entries are zero, and all other entries are independent exponentially distributed random variables with mean 1. Thereby we prove the formula 1+1/4+1/9+\...+1/k2 conjectured by G. Parisi for the case k=m=n, and the generalized conjecture of D. Coppersmith and G. B. Sorkin for arbitrary k, m and n.

Bounds for diluted mean-fields spin glass models

Abstract: In an important recent paper, [2], S. Franz and M. Leone prove rigorous lower bounds for the free energy of the diluted p-spin model and the K-sat model at any temperature. We show that the results for these two models are consequences of a single general principle. Our calculations are significantly simpler than those of [2], even in the replica-symmetric case.

On the rate of convergence in the entropic central limit theorem

Abstract: We study the rate at which entropy is produced by linear combinations of independent random variables which satisfy a spectral gap condition.

Exact L2-small ball behavior of integrated Gaussian processes and spectral asymptotics of boundary value problems

Abstract: We find the exact small deviation asymptotics for the L2-norm of various m-times integrated Gaussian processes closely connected with the Wiener process and the Ornstein – Uhlenbeck process. Using a general approach from the spectral theory of linear differential operators we obtain the two-term spectral asymptotics of eigenvalues in corresponding boundary value problems. This enables us to improve the recent results from [15] on the small ball asymptotics for a class of m-times integrated Wiener processes. Moreover, the exact small ball asymptotics for the m-times integrated Brownian bridge, the m-times integrated Ornstein – Uhlenbeck process and similar processes appear as relatively simple examples illustrating the developed general theory.

Essential spectral radius for Markov semigroups (I): discrete time case

Abstract: Using two new measures of non-compactness βτ(P) and βw(P) for a positive kernel P on a Polish space E, we obtain a new formula of Nussbaum-Gelfand type for the essential spectral radius ress(P) on bℬ. Using that formula we show that different known sufficient conditions for geometric ergodicity such as Doeblin’s condition, drift condition by means of Lyapunov function, geometric recurrence etc lead to variational formulas of the essential spectral radius. All those can be easily transported on the weighted space buℬ. Some related results on L2(μ) are also obtained, especially in the symmetric case. Moreover we prove that for a strongly Feller and topologically transitive Markov kernel, the large deviation principle of Donsker-Varadhan for occupation measures of the associated Markov process holds if and only if the essential spectral radius is zero; this result allows us to show that the sufficient condition of Donsker-Varadhan for the large deviation principle is in fact necessary. The knowledge of ress(P) allows us to estimate eigenvalues of P in L2 in the symmetric case, and to estimate the geometric convergence rate by means of that in the metric of Wasserstein. Applications to different concrete models are provided for illustrating those general results.

Stochastic Loewner evolution in doubly connected domains

Abstract: This paper introduces the annulus SLEκ processes in doubly connected domains. Annulus SLE6 has the same law as stopped radial SLE6, up to a time-change. For κ ≠ 6, some weak equivalence relation exists between annulus SLEκ and radial SLEκ. Annulus SLE2 is the scaling limit of the corresponding loop-erased conditional random walk, which implies that a certain form of SLE2 satisfies the reversibility property. We also consider the disc SLEκ process defined as a limiting case of the annulus SLE’s. Disc SLE6 has the same law as stopped full plane SLE6, up to a time-change. Disc SLE2 is the scaling limit of loop-erased random walk, and is the reversal of radial SLE2.

Random perturbations of dynamical systems and diffusion processes with conservation laws

Abstract: In this paper we consider random perturbations of dynamical systems and diffusion processes with a first integral. We calculate, under some assumptions, the limiting behavior of the slow component of the perturbed system in an appropriate time scale for a general class of perturbations. The phase space of the slow motion is a graph defined by the first integral. This is a natural generalization of the results concerning random perturbations of Hamiltonian systems. Considering diffusion processes as the unperturbed system allows to study the multidimensional case and leads to a new effect: the limiting slow motion can spend non-zero time at some points of the graph. In particular, such delay at the vertices leads to more general gluing conditions. Our approach allows one to obtain new results on singular perturbations of PDE’s.

Random walks in quenched i.i.d. space-time random environment are always a.s. diffusive

Abstract: We consider a general model of discrete-time random walk X t on the lattice ν, ν = 1,..., in a random environment ξ={ξ(t,x):(t,x)∈ ν+1} with i.i.d. components ξ(t,x). Previous results on the a.s. validity of the Central Limit Theorem for the quenched model required a small stochasticity condition. In this paper we show that the result holds provided only that an obvious non-degeneracy condition is met. The proof is based on the analysis of a suitable generating function, which allows to estimate L 2 norms by contour integrals.

Large and moderate deviations for intersection local times

Abstract: We study the large and moderate deviations for intersection local times generated by, respectively, independent Brownian local times and independent local times of symmetric random walks. Our result in the Brownian case generalizes the large deviation principle achieved in Mansmann (1991) for the L 2-norm of Brownian local times, and coincides with the large deviation obtained by Csorgo, Shi and Yor (1991) for self intersection local times of Brownian bridges. Our approach relies on a Feynman-Kac type large deviation for Brownian occupation time, certain localization techniques from Donsker-Varadhan (1975) and Mansmann (1991), and some general methods developed along the line of probability in Banach space. Our treatment in the case of random walks also involves rescaling, spectral representation and invariance principle. The law of the iterated logarithm for intersection local times is given as an application of our deviation results.

measures for branching exit Markov systems and their applications to differential equations

Abstract: Semilinear equations Lu=ψ(u) where L is an elliptic differential operator and ψ is a positive function can be investigated by using (L,ψ)-superdiffusions. In a special case Δu=u2 a powerful probabilistic tool – the Brownian snake – introduced by Le Gall was successfully applied by him and his school to get deep results on solutions of this equation. Some of these results (but not all of them) were extended by Dynkin and Kuznetsov to general equations by applying superprocesses. An important role in the theory of the Brownian snake and its applications is played by measures Open image in new windowx on the space of continuous paths. Our goal is to introduce analogous measures related to superprocesses (and to general branching exit Markov systems). They are defined on the space of measures and we call them Open image in new window-measures. Using Open image in new window-measures allows to combine some advantages of Brownian snakes and of superprocesses as tools for a study of semilinear PDEs.

Vertex-reinforced jump processes on trees and finite graphs

Abstract: We study the continuous time process on the vertices of the b-ary tree which jumps to each nearest neighbor vertex at the rate of the time already spent at that vertex times δ, plus 1, where δ is a positive constant. We show that for fixed b>1, if δ is large enough the process is transient, and if δ is close enough to zero it is recurrent. Related results for some other graphs and trees are also proved.

Diffusion approximation for slow motion in fully coupled averaging

Abstract: In systems which combine fast and slow motions it is usually impossible to study directly corresponding two scale equations and the averaging principle suggests to approximate the slow motion by averaging in fast variables. We consider the averaging setup when both fast and slow motions are diffusion processes depending on each other (fully coupled) and show that there exists a diffusion process which approximates the slow motion in the $L^2$ sense much better than the averaged motion prescribed by the averaging principle.

A convex/log-concave correlation inequality for Gaussian measure and an application to abstract Wiener spaces

Abstract: This paper deals with a generalization of a result due to Brascamp and Lieb which states that in the space of probabilities with log-concave density with respect to a Gaussian measure on this Gaussian measure is the one which has strongest moments. We show that this theorem remains true if we replace x α by a general convex function. Then, we deduce a correlation inequality for convex functions quite better than the one already known. Finally, we prove results concerning stochastic analysis on abstract Wiener spaces through the notion of approximate limit.

Correlations for superpositions and decimations of Laguerre and Jacobi orthogonal matrix ensembles with a parameter

Abstract: A superposition of a matrix ensemble refers to the ensemble constructed from two independent copies of the original, while a decimation refers to the formation of a new ensemble by observing only every second eigenvalue. In the cases of the classical matrix ensembles with orthogonal symmetry, it is known that forming superpositions and decimations gives rise to classical matrix ensembles with unitary and symplectic symmetry. The basic identities expressing these facts can be extended to include a parameter, which in turn provides us with probability density functions which we take as the definition of special parameter dependent matrix ensembles. The parameter dependent ensembles relating to superpositions interpolate between superimposed orthogonal ensembles and a unitary ensemble, while the parameter dependent ensembles relating to decimations interpolate between an orthogonal ensemble with an even number of eigenvalues and a symplectic ensemble of half the number of eigenvalues. By the construction of new families of biorthogonal and skew orthogonal polynomials, we are able to compute the corresponding correlation functions, both in the finite system and in various scaled limits. Specializing back to the cases of orthogonal and symplectic symmetry, we find that our results imply different functional forms to those known previously.

On Lp-theory of stochastic partial differential equations of divergence form in C1 domains

Abstract: Stochastic partial differential equations of divergence form are considered in C1 domains. Existence and uniqueness results are given in a Sobolev space with weights allowing the derivatives of the solutions to blow up near the boundary.

Kernel density estimators: convergence in distribution for weighted sup-norms

Abstract: Let fn denote a kernel density estimator of a bounded continuous density f in the real line. Let Ψ(t) be a positive continuous function such that Open image in new window Under natural smoothness conditions, necessary and sufficient conditions for the sequence Open image in new window \scriptstyle{t\inR/}\big|\Psi(t)(fn(t)-Efn(t))\big| (properly centered and normalized) to converge in distribution to the double exponential law are obtained. The proof is based on Gaussian approximation and a (new) limit theorem for weighted sup-norms of a stationary Gaussian process. This extends well known results of Bickel and Rosenblatt to the case of weighted sup-norms, with the sup taken over the whole line. In addition, all other possible limit distributions of the above sequence are identified (subject to some regularity assumptions).

Large deviations for squares of Bessel and Ornstein-Uhlenbeck processes

Abstract: Let (X t (δ) ,t≥0) be the BESQδ process starting at δx. We are interested in large deviations as ${{\delta \rightarrow \infty}}$ for the family {δ−1 X t (δ) ,t≤T}δ, – or, more generally, for the family of squared radial OUδ process. The main properties of this family allow us to develop three different approaches: an exponential martingale method, a Cramer–type theorem, thanks to a remarkable additivity property, and a Wentzell–Freidlin method, with the help of McKean results on the controlled equation. We also derive large deviations for Bessel bridges.

Sharp bounds on the density, Green function and jumping function of subordinate killed BM

Abstract: Subordination of a killed Brownian motion in a domain D ⊂ R d via an α/2-sta- ble subordinator gives rise to a process Zt whose infinitesimal generator is −(−� |D) α/2 , the fractional power of the negative Dirichlet Laplacian. In this paper we establish upper and lower estimates for the density, Green function and jumping function of Zt when D is either a bounded C 1,1 domain or an exterior C 1,1 domain. Our estimates are sharp in the sense that the upper and lower estimates differ only by a multiplicative constant.

Higher order PDEs and symmetric stable processes

Abstract: We describe a connection between the semigroup of a symmetric stable process with rational index and higher order partial differential equations. As an application, we obtain a variational formula for the eigenvalues associated with the process killed upon leaving a bounded open set D. The variational formula is more ‘‘user friendly’’ than the classical Rayleigh--Ritz formula. We illustrate this by obtaining upper bounds on the eigenvalues in terms of Dirichlet eigenvalues of the Laplacian on D. These results generalize some work of Banuelos and Kulczycki on the Cauchy process. Along the way we prove an operator inequality for the operators associated with the transition densities of Brownian motion and the Brownian motion killed upon leaving D.

The exploration process of inhomogeneous continuum random trees, and an extension of Jeulin’s local time identity

Abstract: We study the inhomogeneous continuum random trees (ICRT) that arise as weak limits of birthday trees. We give a description of the exploration process, a function defined on [0,1] that encodes the structure of an ICRT, and also of its width process, determining the size of layers in order of height. These processes turn out to be transformations of bridges with exchangeable increments, which have already appeared in other ICRT related topics such as stochastic additive coalescence. The results rely on two different constructions of birthday trees from processes with exchangeable increments, on weak convergence arguments, and on general theory on continuum random trees.

Edgeworth expansions of suitably normalized sample mean statistics for atomic Markov chains

Abstract: This paper is devoted to the problem of estimating functionals of type μ(f)=∫fdμ from observations drawn from a positive recurrent atomic Markov chain Open image in new window with stationary distribution μ. The properties of different estimators are studied. Beyond an accurate estimation of their bias, the estimation of their asymptotic variance is considered. We also show that the results of Malinovskii(1987) on the validity of the formal Edgeworth expansion for sample mean statistics of type Tn = n−1∑ni=1f(Xi) extend to their studentized versions, normalized by the asymptotic variance estimates we consider.