The Annals of Probability

Superprocesses and Partial Differential Equations

E. B. Dynkin

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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.

Article information

Source
Ann. Probab., Volume 21, Number 3 (1993), 1185-1262.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176989116

Digital Object Identifier
doi:10.1214/aop/1176989116

Mathematical Reviews number (MathSciNet)
MR1235414

Zentralblatt MATH identifier
0806.60066

JSTOR
links.jstor.org

Subjects
Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60J60: Diffusion processes [See also 58J65] 35K15: Initial value problems for second-order parabolic equations 60J57: Multiplicative functionals 60J65: Brownian motion [See also 58J65] 60J17 35K45: Initial value problems for second-order parabolic systems 60J25: Continuous-time Markov processes on general state spaces

Keywords
Measure-valued processes branching processes branching particle systems super-Brownian motion probabilistic solutions of PDEs nonlinear PDEs graph and range of superdiffusions capacities polar sets Hausdorff measures

Citation

Dynkin, E. B. Superprocesses and Partial Differential Equations. Ann. Probab. 21 (1993), no. 3, 1185--1262. doi:10.1214/aop/1176989116. https://projecteuclid.org/euclid.aop/1176989116


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