Superprocesses and partial differential equations. (English) Zbl 0806.60066

Dynkin, E. B.

Ann. Probab. 21, No. 3, 1185-1262 (1993).

This is a survey of recent developments on measure-valued Markov processes including a number of the author’s important contributions. This subject has its origins in the paper of S. Watanabe [J. Math. Kyoto Univ. 8, 141-167 (1968; Zbl 0159.462)] in which a class of measure-valued branching processes was constructed based on a family of nonlinear semigroups. In recent years the study of these processes has developed in two main directions which are described in Parts I and II, respectively, of this survey.
Part I is devoted to the construction of measure-valued branching processes in a very general setting and in terms of three parameters, a Markov process which describes the spatial motion, an additive functional of this process which defines the intensity of branching and a branching mechanism. The construction of superprocesses is carried out using two different methods, first, as the limit of branching particle systems, and second, directly by the solution of an integral equation which yields the Laplace transforms of the finite-dimensional distributions.
Part II is devoted to the study of superdiffusions and their relations with a class of nonlinear elliptic and parabolic partial differential equations. The graph and range of a superdiffusion are introduced and criteria for \(G\)-polar sets (i.e. subsets of \(S = R_ + \times R^ d\) not hit by the graph of the process), \(R\)-polar sets (i.e. subsets of \(R^ d\) not hit by the range of the process) and a related notion of \(H\)-polarity are derived. This development is based on probabilistic representations of the solutions of the associated nonlinear partial differential equations in terms of superdiffusions and their additive functionals and known analytical results on the pde.
Part III of the paper provides a survey of the historical development and the extensive literature due to many authors which now exists on the subject.
Overall, this survey contains a wealth of information and is essential reading for researchers in this field.

Reviewer: D.A.Dawson (Ottawa)

MSC:

60J80	Branching processes (Galton-Watson, birth-and-death, etc.)
60J60	Diffusion processes
35K15	Initial value problems for second-order parabolic equations
60-02	Research exposition (monographs, survey articles) pertaining to probability theory
60G57	Random measures
60J65	Brownian motion
35K45	Initial value problems for second-order parabolic systems
60J25	Continuous-time Markov processes on general state spaces
35-02	Research exposition (monographs, survey articles) pertaining to partial differential equations

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; survey; nonlinear semigroups

Citations:

Zbl 0159.462