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:**

- 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*