Integration by parts on δ-bessel bridges, δ > 3 and related SPDEs

• 成果类型: Article
• 署名作者: Zambotti, L
• 署名单位: Scuola Normale Superiore di Pisa
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• 发表日期: 2003
• 页码: 323-348
• 关键词: heat-equation; reflection

摘要

We study a white-noise driven semilinear partial differential equation on the spatial interval [0, 1] with Dirichlet boundary condition and with a singular drift of the form cu(-3), c > 0. We prove existence and uniqueness of a non-negative continuous adapted solution u on [0, infinity) x [0, 1] for every nonnegative continuous initial datum x, satisfying x(0) = x(1) = 0. We prove that the law pi(delta) of the Bessel bridge on [0, 1] of dimension delta > 3 is the unique invariant probability measure of the process x --> u, with c = (delta - 1)(delta - 3)/8 and, if delta is an element of N, that u is the radial part in the sense of Dirichlet forms of the R-delta-valued solution of a linear stochastic heat equation. An explicit integration by parts formula w.r.t. pi(delta) is given for all delta > 3.