Diffusions and Their Applications

University Of Connecticut, Storrs CT

Investigators

Abstract

Several problems in probability theory will be studied. The first is the question of uniqueness for diffusions on fractals. For infinitely ramified fractals such as the Sierpinski carpet it is not yet known whether all possible constructions of Brownian motion whose state space is the fractal lead to the same process. This problem has applications to mathematical physics. Another concerns applying probabilistic techniques to the hot spots problem, a difficult problem in analysis. The hot spots problem is to determine where the solution to the heat equation with reflecting boundaries takes its maximum. A third problem is the estimation of the heat kernel, a problem that is related to partial differential equations. The heat equation for elliptic operators in divergence form has a fundamental solution that is comparable to that for the Laplacian. Is this true when the state space is not Euclidean space, but rather a more general manifold? Several problems in probability theory will be studied. They are all related to analysis and partial differential equations, in particular, to the heat equation. The heat equation governs the flow of heat as well as many other physical quantities. Interestingly enough, the study of random processes such as Brownian motion, can shed a great deal of light on the solutions of the heat equation in various media.