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

• 成果类型: Article
• 署名作者: Fournier, N
• 署名单位: Universite de Lorraine
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• 发表日期: 2002
• 页码: 135-170
• 关键词: Existence; REGULARITY; calculus; points

摘要

Consider the one-dimensional solution X = {X-t}(tis an element of[0,T]) of a possibly degenerate stochastic differential equation driven by a (non compensated) Poisson measure. We denote by M a set of deterministic integer-valued measures associated with the considered Poisson measure. For in m is an element of M, we denote by S(m) = {S-t(m)}(tis an element of[0,T]) the skeleton associated with X. We assume some regularity conditions, which allow to define a sort of derivative DSt (m) of S-t (m) with respect to m. Then we fix t is an element of]0, T], y is an element of R, and we prove that as soon there exists in m is an element of M such that S-t(m) = y, DSt(m) not equal 0 and DeltaS(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.