Nonlinear parabolic PDE and additive functionals of superdiffusions

• 成果类型: Article
• 署名作者: Dynkin, EB; Kuznetsov, SE
• 署名单位: Cornell University; Russian Academy of Sciences
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• 发表日期: 1997
• 页码: 662-701
• 关键词: partial-differential equations; branching-processes; superprocesses; SINGULARITIES; REGULARITY

摘要

Suppose that E is an arbitrary domain in R-d, L is a second order elliptic differential operator in S = R+ x E and S-e is the extremal part of the Martin boundary for the corresponding diffusion xi. Let 1 < alpha less than or equal to 2. We investigate a boundary value problem partial derivative u/partial derivative r + Lu - u(alpha) = -eta inS, (*) u = v on S-e, u = 0 on {infinity} x E involving two measures eta and v. For the existence of a solution, we give sufficient conditions in terms of a Martin capacity and necessary conditions in terms of hitting probabilities for an (L, alpha)-superdiffusion X. If a solution exists, then it can be expressed by an explicit formula through an additive functional A of X. An (L, alpha)-superdiffusion is a branching measure-valued process. A natural linear additive (NLA) functional A of X is determined uniquely by its potential h defined by the formula P(mu)A(0, infinity) = integral h(r, x)mu(dr, dx) for all mu is an element of M* (the determining set of A). Every potential h is an exit rule for xi and it has a unique decomposition into extremal exit rules. If eta and v, are measures which appear in this decomposition, then (*) can be replaced by an integral equation (**) u(r, x) + integral p(r, x;t, dy)u(t, y)(alpha) ds = h(r, x), where p(r, x; t, dy) is the transition function of xi. We prove that h is the potential of a NLA functional if and only if(**) has a solution u. Moreover, u(r, x) = -logP(r, x)e(-A(0, infinity)). By applying these results to homogeneous functionals of time-homogeneous superdiffusions, we get a stronger version of theorems proved in an earlier publication. The foundation for our present investigation is laid by a general theory developed in the accompanying paper.