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

The subject of this article is a class of measure-valued Markov proces ses. A typical example is super-Brownian motion . The Laplacian DELTA plays a fundamental role in the theory of Brownian motion. For super-B rownian motion, an analogous role is played by the operator DELTAu - p si(u), where a nonlinear function psi describes the branching mechanis m. The class of admissible functions psi includes the family psi(u) = u(alpha), 1 < alpha less-than-or-equal-to 2. Super-Brownian motion bel ongs to the class of continuous state branching processes investigated in 1968 in a pioneering work of Watanabe. Path properties of super-Br ownian motion are well known due to the work of Dawson, Perkins, Le Ga ll and others. Partial differential equations involving the operator D ELTAu - psi(u) have been studied independently by several analysts, in cluding Loewner and Nirenberg, Friedman, Brezis, Veron, Baras and Pier re. Connections between the probabilistic and analytic theories have b een established recently by the author.