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

CNRS U.M.R. 7599
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``Probabilités et Modèles Aléatoires''
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**Auteur(s): **

**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*