Superprocesses and Partial Differential Equations

Abstract

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 $\Delta u - \psi(u)$, where a nonlinear function $\psi$ describes the branching mechanism. The class of admissible functions $\psi$ includes the family $\psi(u) = u^\alpha, 1 < \alpha \leq 2$. 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 $\Delta u - \psi(u)$ 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.