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

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

Existence results for 2D homogeneous Boltzmann equations without cutoff and for non Maxwell molecules by use of Malliavin calculus


Code(s) de Classification MSC:

Résumé: Tanaka showed in [21] a way to relate the measure-solution $\{Q_t\}_t$ of a spatially homogeneous Boltzmann equation of Maxwell molecules without angular cutoff to the solution $V_t$ of a Poisson-driven stochastic differential equation: for each $t$, $Q_t$ is the law of $V_t$.\\ Using a typically probabilistic substitution in the Boltzmann equation,we extend Tanaka's probabilistic interpretation to much more general spatially homogeneous Boltzmann equations. \\ Then we introduce an adapted stochastic calculus of variations on the Poisson space, to prove that for each $t>0$, the law of $V_t$ admits a density $f(t,.)$. The function $f(t,v)$ is solution of the Boltzmann equation, and this existence result improves the existing analytical results. \\ Since the "Malliavin derivative" of $V_t$ does not belong to $L^2(\Omega)$, and thus cannot be a "$L^2$-derivative", we introduce a criterion of absolute continuity based on the use of a.s. derivatives.

Mots Clés: Boltzmann equations without cutoff ; Nonlinear stochastic differential equations ; Jump measures ; Stochastic calculus of variations

Date: 2000-11-20

Prépublication numéro: PMA-622