• H. GUÉRIN
• S. MÉLÉARD
• E. NUALART

Code(s) de Classification MSC:

• 60J75 Jump processes
• 60H10 Stochastic ordinary differential equations [See also 34F05]
• 60K35 Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
• 82C40 Kinetic theory of gases

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