Université Paris 6
Pierre et Marie Curie
Université Paris 7
Denis Diderot

CNRS U.M.R. 7599
``Probabilités et Modèles Aléatoires''

Exponential estimates for spatially homogeneous Landau equations via the Malliavin calculus


Code(s) de Classification MSC:

Résumé: The aim of this paper is to show how a probabilistic approach and the use of Malliavin calculus provide exponential estimates for the solution of a spatially homogeneous Landau equation, for a generalization of Maxwellian molecules. We recall how this solution can be obtained as the density of a nonlinear process. This process is a diffusion driven by a space-time white noise, with linear growth, but unbounded coefficients, and a degenerate diffusion matrix. However, the nonlinearity gives some non-degeneracy which implies the existence and regularity of the density. We use some ideas introduced by A. Kohatsu-Higa and developed by V. Bally, adapted to our situation to show that this density can be upper and lower bounded by some exponential-type estimates.

Mots Clés: Spatially homogeneous Landau equation ; Nonlinear stochastic differential equations ; Malliavin calculus ; exponential estimates

Date: 2004-01-26

Prépublication numéro: PMA-876