Alain-Sol Sznitman

Équations de type de Boltzmann, spatialement homogènes

SummaryWe study the construction of a nonhomogeneous ℝn-valued Markov process, whose laws at fixed times evolve according to a Boltzmann type equation (spatially homogeneous). We then consider the problem of the asymptotic propagation of chaos for a system of a large number of interacting particles.

Nonlinear reflecting diffusion process, and the propagation of chaos and fluctuations associated

Abstract A nonlinear diffusion satisfying a normal reflecting boundary condition is constructed and a result of propagation of chaos for a system of interacting diffusing particles with normal reflecting boundary conditions is proven. Then a gaussian limit for the fluctuation field which is defined in L 0 2 ( B ) of a Wiener type space B is obtained. The covariance of the gaussian limit is computed in terms of a Hilbert-Schmidt operator on L 0 2 ( B ).

Ten lectures on random media

The field of random media has been the object of intensive activity over the last twenty-five years. It gathers a variety of models generally originating from physical sciences, where certain materials or substances have defects or inhomogeneities. This feature can be taken into account by letting the medium be random. Randomness in the medium turns out to cause very unexpected effects, especially in the large-scale behavior of some of these models. What in the beginning was often deemed to be a simple toy-model ended up as a major mathematical challenge. After more than twenty years of intensive research in this field, certain new paradigms and some general methods have emerged, and the surprising results on the asymptotic behavior of individual models are now better understood in more general frameworks. This monograph grew out of the DMV Lectures on Random Media held by the authors at the Mathematical Research Institute in Oberwolfach in November 1999 and tries to give an account of some of the developments in the field, especially in the area of random motions in random media and of mean-field spin glasses. It will be a valuable resource for postgraduates and researchers in probability theory and mathematical physics.

Cut points and diffusive random walks in random environment

We study in this work a special class of multidimensional random walks in random environment for which we are able to prove in a non-perturbative fashion both a law of large numbers and a functional central limit theorem. As an application we provide new examples of diffusive random walks in random environment. In particular we construct examples of diffusive walks which evolve in an environment for which the static expectation of the drift does not vanish.

Giant component and vacant set for random walk on a discrete torus

We consider random walk on a discrete torus

Brownian asymptotics in a Poissonian environment

E

Upper bound on the disconnection time of discrete cylinders and random interlacements

of side-length

Martingales dépendant d'un paramètre: une formule d'Ito

N

Lifschitz tail and wiener sausage. I

, in sufficiently high dimension

How universal are asymptotics of disconnection times in discrete cylinders

d