Université Paris 6Pierre et Marie Curie Université Paris 7Denis Diderot CNRS U.M.R. 7599 Probabilités et Modèles Aléatoires''

### Strict positivity of the density for simple jump processes using the tools of support theorems. Application to the Kac equation without cutoff.

Auteur(s):

Code(s) de Classification MSC:

• 60H07 Stochastic calculus of variations and the Malliavin calculus
• 60H10 Stochastic ordinary differential equations, See Also {
• 60J75 Jump processes
• 82C40 Kinetic theory of gases

Résumé: Consider the one-dimensional solution $X=\{X_t\}_{t \in [0,T]}$ of a possibly degenerate stochastic differential equation driven by a (non compensated) Poisson measure. We denote by $\cm$ a set of deterministic integer-valued measures associated with the considered Poisson measure. For $m\in \cm$, we denote by $S(m)=\{S_t(m)\}_{t\in[0,T]}$ the skeleton associated with $X$. We assume some regularity conditions, which allow to define a sort of "derivative" $D S_t(m)$ of $S_t(m)$ with respect to $m$. Then we fix $t \in ]0,T]$, $y\in \reel$, and we prove that as soon there exists $m\in \cm$ such that $S_t(m)=y$, $DS_t(m) \ne 0$, and $\Delta S_t(m) =0$, the law of $X_t$ is bounded below by a nonnegative measure admitting a continuous density not vanishing at $y$. In the case where the law of $X_t$ admits a continuous density $p_t$, this means that $p_t(y)>0$. We finally apply the described method in order to prove that the solution to a Kac equation without cutoff does never vanish.

Mots Clés: Stochastic differential equations with jumps ; Stochastic calculus of variations ; Support theorems ; Boltzmann equations

Date: 2000-03-30

Prépublication numéro: PMA-581