Diffusions, Superdiffusions and Partial Differential Equations

线性椭圆型和抛物型偏微分方程与马尔可夫扩散过程之间的密切关系在上个世纪这两个领域的发展中发挥了重要作用。在过去15年左右的时间里，这一方向的新篇章——一类半线性方程和超扩散之间的相互作用——已经发展起来。这一理论已达到一定的成熟阶段。然而，一些关键问题仍然没有解决（例如，给定细迹解的唯一性和超扩散出口边界的结构）。关于这些问题的工作是提案的主题。超扩散是一类特殊的分支测量值过程，也称为超过程。在纯数学之外，它们在群体遗传学中发挥着重要作用，并为研究具有无限多个自由度的复杂物理系统提供了新的工具。大约二十年来，这些过程吸引了世界各地许多研究者的努力。

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.