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年中，对这两类之间联系的深入研究导致了与布朗运动有关的经典概率势理论的非线性模拟。在过去的三年中，该领域的一些基本问题得到了解决，开辟了新的研究方向。我们将对这些方向进行探讨。特别是，新的概率工具将应用于非线性微分方程的问题。研究了与超过程相关的测度空间中的调和函数，有望为无限维分析开辟新的篇章。在这方面，超扩散的出口边界将被探索-尝试将马丁边界理论扩展到非线性偏微分方程。该提案的目的是为概率分析做出贡献，这是现代数学的一个重要分支。随机过程理论与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。起初，大多数分析结果是由概率学家使用的。最近，分析师（和物理学家）从概率方法中获得了灵感。当然，分析学的发展，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.