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:**

- 60H30 Applications of stochastic analysis (to PDE, etc.)
- 60H07 Stochastic calculus of variations and the Malliavin calculus
- 82C40 Kinetic theory of gases

**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*