Diffusions, Superdiffusions and Partial Differential Equations

Cornell University, Ithaca NY

Investigators

Abstract

A close relationship between linear elliptic and parabolic partial differential equations and Markov diffusion processes played an important role in the development of both fields during the last century. A new chapter in this direction -- an interplay between a class of semilinear equations and superdiffusions -- has evolved over the past 15 or so years. The theory has reached a certain stage of maturity. However some key problems remain open (for instance, the uniqueness of the solution with a given fine trace and the structure of exit boundaries of superdiffusions). The work on these problems is the subject of the proposal. Superdiffusions is a special class of branching measure-valued processes also known as superprocesses. Outside of pure mathematics, they play a significant role in population genetics and they provide new tools for the study of complex physical systems with infinitely many degrees of freedom. For about two decades these processes have attracted the efforts of many investigators around the world.