Diffusion Processes and Stochastic Analysis

University Of Washington, Seattle WA

Investigators

Abstract

The PIs study several problems in stochastic analysis. The first problem involves reflected Brownian motion in time-dependent domains and the corresponding heat equation with the Neumann boundary conditions. The main emphasis is on the existence and uniqueness of the solutions to the heat equation and the reflected Brownian motion, and on the singularities of the heat equation solutions close to the moving boundary. The second part of the project is concerned with stochastic flows related to singular stochastic differential equations. Flows of this type have interesting combinatorial properties not present in flows corresponding to equations with smooth coefficients. The "hot spot" conjecture states that hottest point in an insulated body lies on its boundary. While this is not true in general, it is a widespread belief that the conjecture holds in convex domains. The PIs are currently studying symmetric convex domains. Finally, stable processes and related processes are studied from the point of view of potential theory. Stochastic analysis was one of the most important developments on the borderline of probability and analysis in the twentieth century. It now provides the basis of studying fundamental properties of real life phenomena which are random by nature. One of the most spectacular recent successes of the theory is the so-called financial mathematics. This theory provides a solid theoretical basis for trading securities - its founding fathers were recently recognized by a Nobel Prize in Economics. Stable processes, one of the topics of the project, are more and more often applied in financial mathematics and other applied sciences because the traditional continuous models are not always adequate. The study of a singular flow was directly inspired by a collaboration of one of the PIs (Burdzy) with economists, published in a leading journal "Econometrics." The study of reflected Brownian motion in time dependent domains is reminiscent of the "Stefan problem" concerned with melting ice. Similar physical models are of great interest to scientists who model real life environmental changes.