THE 1991 WALD MEMORIAL LECTURES - SUPERPROCESSES AND PARTIAL-DIFFERENTIAL EQUATIONS

• 成果类型: Article
• 署名作者: DYNKIN, EB
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/aop/1176989116
• 发表日期: 1993
• 页码: 1185-1262
• 关键词: valued branching-processes; super-brownian motion; nonlinear elliptic-equations; semilinear heat-equations; singular solutions; blow-up; REPRESENTATION; functionals; absorption; DIFFUSIONS

摘要

The subject of this article is a class of measure-valued Markov processes. A typical example is super-Brownian motion . The Laplacian DELTA plays a fundamental role in the theory of Brownian motion. For super-Brownian motion, an analogous role is played by the operator DELTAu - psi(u), where a nonlinear function psi describes the branching mechanism. The class of admissible functions psi includes the family psi(u) = u(alpha), 1 < alpha less-than-or-equal-to 2. Super-Brownian motion belongs to the class of continuous state branching processes investigated in 1968 in a pioneering work of Watanabe. Path properties of super-Brownian motion are well known due to the work of Dawson, Perkins, Le Gall and others. Partial differential equations involving the operator DELTAu - psi(u) have been studied independently by several analysts, including Loewner and Nirenberg, Friedman, Brezis, Veron, Baras and Pierre. Connections between the probabilistic and analytic theories have been established recently by the author.