Lower bounds for the density of locally elliptic Ito processes

• 成果类型: Article
• 署名作者: Bally, Vlad
• 署名单位: Universite Gustave-Eiffel; Universite Paris-Est-Creteil-Val-de-Marne (UPEC); Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI)
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/009117906000000458
• 发表日期: 2006
• 页码: 2406-2440
• 关键词: fundamental-solutions; spdes

摘要

We give lower bounds for the density p(T) (x, y) of the law of X-t, the solution of dX(t) = sigma(X-t)dB(t) +b(X-t)dt, X-0 = x, under the following local ellipticity hypothesis: there exists a deterministic differentiable curve x(t), 0 <= t <= T, such that x(0) = x, x(T) = y and sigma sigma* (x(t)) > 0, for all t is an element of [0, T]. The lower bound is expressed in terms of a distance related to the skeleton of the diffusion process. This distance appears when we optimize over all the curves which verify the above ellipticity assumption. The arguments which lead to the above result work in a general context which includes a large class of Wiener functionals, for example, Ito processes. Our starting point is work of Kohatsu-Higa which presents a general framework including stochastic PDE's.