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January 2002 Strict Positivity of the Density for Simple Jump Processes Using the Tools of Support Theorems. Application to the KAC Equation Without Cutoff
Nicolas Fournier
Ann. Probab. 30(1): 135-170 (January 2002). DOI: 10.1214/aop/1020107763

Abstract

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 $\mathcal{M}$ a set of deterministic integer-valued measures associated with the considered Poisson measure. For $m\in \mathcal{M}$, 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 $m\in \mathcal{M}$ 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.

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Nicolas Fournier. "Strict Positivity of the Density for Simple Jump Processes Using the Tools of Support Theorems. Application to the KAC Equation Without Cutoff." Ann. Probab. 30 (1) 135 - 170, January 2002. https://doi.org/10.1214/aop/1020107763

Information

Published: January 2002
First available in Project Euclid: 29 April 2002

zbMATH: 1017.60063
Digital Object Identifier: 10.1214/aop/1020107763

Subjects:
Primary: 60H07 , 60H10 , 60J75 , 82C40

Keywords: Boltzmann equations , stochastic calculus of variations , Stochastic differential equations with jumps , support theorems

Rights: Copyright © 2002 Institute of Mathematical Statistics

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Vol.30 • No. 1 • January 2002
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