Natural linear additive functionals of superprocesses

• 成果类型: Article
• 署名作者: Dynkin, EB; Kuznetsov, SE
• 署名单位: Russian Academy of Sciences; Central Economics & Mathematics Institute RAS
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• 发表日期: 1997
• 页码: 640-661
• 关键词: partial-differential equations; branching-processes; superdiffusions

摘要

We investigate natural linear additive (NLA) functionals of a general critical (xi, K, psi)-superprocess X. We prove that all of them have only fixed discontinuities. All homogeneous NLA functionals of time-homogeneous superprocesses are continuous (this was known before only in the case of quadratic branching). We introduce an operator epsilon(u) defined in terms of (xi, K, psi) and we prove that the potential h and the log-potential u of a NLA functional A are connected by the equation u + epsilon(u) = h. The potential is always an exit rule for xi and the condition h + epsilon(h) < infinity a.e. is sufficient for an exit rule h to be a potential. In an accompanying paper, these results are applied to boundary value problems for partial differential equations involving nonlinear operator Lu - u(alpha) where L is a second order elliptic differential operator and 1 < alpha less than or equal to 2.