On the concentration of Sinai’s walk

We consider Sinai’s random walk in a random environment. We prove that for an interval of time [1,n]$[1,n]$ Sinai’s walk sojourns in a small neighborhood of the point of localization for the quasi-totality of this amount of time. Moreover the local time at the point of localization normalized by n$n$ converges in probability to a well defined random variable of the environment. From these results we get applications to the favorite sites of the walk and to the maximum of the local time.     

Lévy processes with no positive jumps at an increase time

Summary We study the behaviour of a Lévy process with no positive jumps near its increase times. Specifically, we construct a local time on the set of increase times. Then, we describe the path decomposition at an increase time chosen at random according to the local time, and we evaluate the rate of escape before and after this instant.     

Brownian motion on a homogeneous random fractal

Summary We introduce a simple random fractal based on the Sierpinski gasket and construct a Brownian motion upon the fractal. The properties of the process on the Sierpinski gasket are modified by the random environment. A sample path construction of the process via time truncation is used, which is a direct construction of the process on the fractal from the associated Dirichlet forms. We obtain estimates on the resolvent and transition density for the process and hence a value for the spectral dimension which satisfiesd _s=2d _f/d_w. A branching process in a random environment can be used to deduce some of the sample path properties of the process.     

Gene flow across geographical barriers — scaling limits of random walks with obstacles

We study a class of random walks which behave like simple random walks outside of a bounded region around the origin and which are subject to a partial reflection near the origin. We obtain a non trivial scaling limit which behaves like reflected Brownian motion until its local time at zero reaches an exponential variable. It then follows reflected Brownian motion on the other side of the origin until its local time at zero reaches another exponential level, etc. These random walks are used in population genetics to trace the position of ancestors in the past near geographical barriers.     

Cyclically stationary Brownian local time processes

Summary. Local time processes parameterized by a circle, defined by the occupation density up to time T of Brownian motion with constant drift on the circle, are studied for various random times T. While such processes are typically non-Markovian, their Laplace functionals are expressed by series formulae related to similar formulae for the Markovian local time processes subject to the Ray–Knight theorems for BM on the line, and for squares of Bessel processes and their bridges. For T the time that BM on the circle first returns to its starting point after a complete loop around the circle, the local time process is cyclically stationary, with same two-dimensional distributions, but not the same three-dimensional distributions, as the sum of squares of two i.i.d. cyclically stationary Gaussian processes. This local time process is the infinitely divisible sum of a Poisson point process of local time processes derived from Brownian excursions. The corresponding intensity measure on path space, and similar Lévy measures derived from squares of Bessel processes, are described in terms of a 4-dimensional Bessel bridge by Williams’ decomposition of It?’s law of Brownian excursions. Received: 28 June 1995     

Valleys and the Maximum Local Time for Random Walk in Random Environment

Let ξ (n, x) be the local time at x for a recurrent one-dimensional random walk in random environment after n steps, and consider the maximum ξ^*(n) = max _x ξ(n, x). It is known that lim sup is a positive constant a.s. We prove that lim inf is a positive constant a.s. this answers a question of P. Révész [5]. The proof is based on an analysis of the valleys in the environment, defined as the potential wells of record depth. In particular, we show that almost surely, at any time n large enough, the random walker has spent almost all of its lifetime in the two deepest valleys of the environment it has encountered. We also prove a uniform exponential tail bound for the ratio of the expected total occupation time of a valley and the expected local time at its bottom.     

Reflected Brownian motion in generic triangles and wedges

Consider a generic triangle in the upper half of the complex plane with one side on the real line. This paper presents a tailored construction of a discrete random walk whose continuum limit is a Brownian motion in the triangle, reflected instantaneously on the left and right sides with constant reflection angles. Starting from the top of the triangle, it is evident from the construction that the reflected Brownian motion lands with the uniform distribution on the base. This raises some questions on the possible distributions of hulls generated by local processes.     

Some extensions of the arc sine law as partial consequences of the scaling property of Brownian motion

Summary The scaling property of Brownian motion is exploited systematically in order to extend Paul Lévy's arc sine law to Brownian motion perturbed by its local time at 0. Other important ingredients of the proofs are some Ray-Knight theorems for local times.     

The law of iterated logarithm for additive functionals and martingale additive functionals of Harris recurrent Markov processes

We use Nummelin splitting in continuous time in order to prove laws of iterated logarithm for additive functionals of a Harris recurrent Markov process, with deterministic or random renormalization.     

Interacting Fisher–Wright diffusions in a catalytic medium

We study the longtime behaviour of interacting systems in a randomly fluctuating (space–time) medium and focus on models from population genetics. There are two prototypes of spatial models in population genetics: spatial branching processes and interacting Fisher–Wright diffusions. Quite a bit is known on spatial branching processes where the local branching rate is proportional to a random environment (catalytic medium). Here we introduce a model of interacting Fisher–Wright diffusions where the local resampling rate (or genetic drift) is proportional to a catalytic medium. For a particular choice of the medium, we investigate the longtime behaviour in the case of nearest neighbour migration on the d-dimensional lattice. While in classical homogeneous systems the longtime behaviour exhibits a dichotomy along the transience/recurrence properties of the migration, now a more complicated behaviour arises. It turns out that resampling models in catalytic media show phenomena that are new even compared with branching in catalytic medium. Received: 15 November 1999 / Revised version: 16 June 2000 / Published online: 6 April 2001     

Asymptotically one-dimensional diffusions on the Sierpinski gasket and theabc-gaskets

Summary Diffusion processes on the Sierpinski gasket and theabc-gaskets are constructed as limits of random walks. In terms of the associated renormalization group, the present method uses the inverse trajectories which converge to unstable fixed points corresponding to the random walks on one-dimensional chains. In particular, non-degenerate fixed points are unnecessary for the construction. A limit theorem related to the discrete-time multi-type non-stationary branching processes is applied.     

On the local time of random walk on the 2-dimensional comb

We study the path behaviour of general random walks, and that of their local times, on the 2-dimensional comb lattice C² that is obtained from Z² by removing all horizontal edges off the x-axis. We prove strong approximation results for such random walks and also for their local times. Concentrating mainly on the latter, we establish strong and weak limit theorems, including Strassen-type laws of the iterated logarithm, Hirsch-type laws, and weak convergence results in terms of functional convergence in distribution.     

On the most visited sites by a symmetric stable process

Summary. At time t, the most visited site of a linear Brownian motion is defined as the point which realises the supremum of the local times at time t. Let V be the time indexed process of the most visited sites by a linear Brownian motion. We show that every value is polar for V. Those results are extended from Brownian motion to symmetric stable processes, and then to the absolute value of a symmetric stable process. Received: 1 March 1996 / In revised form: 17 October 1996     

An invariance principle for non-symmetric Markov processes and reflecting diffusions in random domains

Summary We study an invariance principle for additive functionals of nonsymmetric Markov processes with singular mean forward velocities. We generalize results of Kipnis and Varadhan [KV] and De Masi et al. [De] in two directions: Markov processes are non-symmetric, and mean forward velocities are distributions. We study continuous time Markov processes. We use our result to homogenize non-symmetric reflecting diffusions in random domains.     

Intersection local times for interlacements

We define renormalized intersection local times for random interlacements of Lévy processes in R^d$R^d$ and prove an isomorphism theorem relating renormalized intersection local times with associated Wick polynomials.     

Local times on curves and uniform invariance principles

Summary Sufficient conditions are given for a family of local times |L _t ^µ | ofd-dimensional Brownian motion to be jointly continuous as a function oft and . Then invariance principles are given for the weak convergence of local times of lattice valued random walks to the local times of Brownian motion, uniformly over a large family of measures. Applications included some new results for intersection local times for Brownian motions on ² and ².Research partially supported by NSF grant DMS-8822053     

The problem of the most visited site in random environment

We prove that the process of the most visited site of Sinai's simple random walk in random environment is transient. The rate of escape is characterized via an integral criterion. Our method also applies to a class of recurrent diffusion processes with random potentials. It is interesting to note that the corresponding problem for the usual symmetric Bernoulli walk or for Brownian motion remains open. Received: 17 April 1998     

Dynamics for the Brownian web and the erosion flow

The Brownian web is a random object that occurs as the scaling limit of an infinite system of coalescing random walks. Perturbing this system of random walks by, independently at each point in space–time, resampling the random walk increments, leads to some natural dynamics. In this paper we consider the corresponding dynamics for the Brownian web. In particular, pairs of coupled Brownian webs are studied, where the second web is obtained from the first by perturbing according to these dynamics. A stochastic flow of kernels, which we call the erosion flow, is obtained via a filtering construction from such coupled Brownian webs, and the N$N$-point motions of this flow of kernels are identified.     

Local times of stochastic processes with positive definite bivariate densities

A new class of stochastic processes, called processes of positive bivariate type, is defined. Such a process is typically one whose bivariate density functions are positive definite, at least for pairs of time points which are sufficiently mutually close. The class includes stationary Gaussian processes and stationary reversible Markov processes, and is closed under the operations of composition and convolution. The purpose of this work is to show that the local times of such processes can be investigated in a natural way. One of the main contributions is an orthogonal expansion of the local time which is new even in the well-studied stationary Gaussian case. The basic tool in its construction is the Lancaster-Sarmanov expansion of a bivariate density in a series of canonical correlations and canonical variables.     

Mean first passage times of two-dimensional processes with jumps

In this paper, we incorporate a jump component into the model based on a two-dimensional degenerate diffusion process for the remaining lifetime of machines in the recent paper [Lefebvre, M., 2010. Mean first-passage time to zero for wear processes. Stochastic Models 26, 46-53] by the second author. We calculate explicitly the expected value of first passage times associated to the two-dimensional process when the jump component is taken to be a compound Poisson process with exponential jumps and random proportion of jumps.