Superdiffusions and Partial Differential Equations

该项目的主题是一类称为超扩散的测度值马尔可夫过程和相应的半线性椭圆方程。 在过去的15年里，对这两类之间的联系进行了深入的研究，结果开发了与布朗运动相关的经典概率势理论的非线性模拟。 在过去的三年里，这一领域的一些基本问题得到了解决，开辟了新的研究方向。 将探讨这些方向。 特别是，新的概率工具将应用于非线性微分方程的问题。 超过程测度空间中的调和函数将被研究，这将为无限维分析开辟新的篇章。 在这方面，出口边界的超扩散将探讨-试图扩展马丁边界理论的非线性偏微分方程。该建议的目标是有助于概率分析，这是一个重要的分支的现代数学。 随机过程理论和偏微分方程理论之间的相互作用对这两个领域都是有益的。 最初，大多数分析结果被概率学家使用。 最近，分析师（和物理学家）从概率方法中获得了灵感。 当然，一般分析的发展，特别是微分方程理论的发展，在很大程度上是由物理问题推动的。 物理学和概率论的区别在于，后者不仅提供了一种直觉，而且提供了证明定理的严格工具。

The subject of the project is a class of measure-valued Markov processes called superdiffusions and the correponding class of semilinear elliptic equations. An intensive study of the connections between these two classes during the past 15 years resulted in developing a nonlinear analog of the classical probabilistic potential theory related to the Brownian motion. Some fundamental problems in this field were solved during the last three years which opens new directions of research. These directions will be explored. In particular, new probabilistic tools will be applied to problems on nonlinear differential equations. Harmonic functions in spaces of measures associated with superprocesses will be investigated which promises to open a new chapter in infinite dimensional analysis. In this connection, exit boundaries for superdiffusions will be explored - an attempt to extend Martin boundary theory to nonlinear PDEs.The goal of the proposal is to contribute to probabilistic analysis, which is an important branch of modern mathematics. Interactions between the theory of stochastic processes and the theory of partial differential equations are beneficial for both fields. At the beginning, mostly analytic results were used by probabilists. More recently, analysts (and physicists) took inspiration from the probabilistic approach. Of course, the development of analysis in general, and of theory of differential equations in particular, was motivated to a great extent by problems in physics. A difference between physics and probability is that the latter provides not only an intuition but also rigorous tools for proving theorems.