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.
