Mathematical Sciences: Branching Measure-Valued Processes and Related Nonlinear Partial Differential Equations

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9623190 Dynkin ABSTRACT Investigations on branching measure-valued processes and their connections with nonlinear PDEs are proposed. One of the targets is the boundary theory of superdiffusion on a differentiable manifold and related problems on the equation Lu=F(u) where L is the second order elliptic differential operator and F is a nonlinear function. Among the problems are the description of all positive solutions and their behavior at the Martin boundary, and criteria for removable boundary singularities in terms of the Martin capacity. This part of the project has been prepared for by an intensive study, done in the last years, of the case of a bounded domain with a smooth boundary in a Euclidean space. Another direction iq the investigation of the structure of the most general branching measure-valued process. Previous results have been obtained under an assumption that the second moment of the total mass is finite. There is a hope to remove or to weaken this restriction by using recent progress in the theory of linear additive functionals. During the last decade branching measure-valued processes also known as superprocesses have attracted a great deal of effort from many investigators around the world. Interest in these processes is motivated by their applications to biology, especially, to population genetics. On the other hand, this class of processes provides new tools for the study of complex physical systems with infinitely many degrees of freedom.