RUI: Problems in Stochastic Geometry

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The investigator plans to study three problems in the area of stochasticdifferential geometry. The first problem concerns the Gauss-Bonnet-Cherntheorem. The investigator has proved a generalization of this theorem validfor an oriented even-dimensional Riemannian vector bundle over a closedcompact manifold, equipped with a metric connection. The investigator wouldlike to simplify his proof, replacing the use of the Splitting Principle,which constitutes a key step in his argument, with a more elementary andtransparent stochastic argument. The second problem concerns the derivationof integration by parts formulae for the law of a diffusion process on acompact manifold. The investigator has recently found a new method forobtaining integration by parts formulae in the case where the diffusion isstrictly elliptic. He plans to use his method to study the analogous problemfor degenerate (non-elliptic) diffusions. He believes he will be able toprove a dichotomy theorem giving conditions very close to necessary andsufficient for the existence of integration by parts formulae, for a largeclass of vector fields in the degenerate case. In the third problem, theinvestigator in collaboration with Elton Hsu, will use ideas in two papersof Hsu to give a simplified stochastic proof of the Gauss-Bonnet-Cherntheorem for manifolds with boundary. They will use the expertise they gainfrom this work to study the index theorem for the Dirac operator on spinmanifolds with boundary. The proposal combines two areas of mathematics, stochastic analysis, thestudy of randomness, and differential geometry, the study of shape. Bothareas have close connections with the physical world. For example,stochastic differential equations, a central theme of the proposal, arewidely used to model physical systems subject to the influence of randomnoise. Examples include the flow of heat in a material, weather systems,the trajectory of a spacecraft, and the pricing of stock options. Curvedspaces are often the natural setting for these phenomena, e.g. if one wishesto study the heat at various points on a cylindrical pipe. A deeperunderstanding of the mathematics underlying physical systems is oftencrucial in developing good mathematical models of these systems. Thus,although it is of a theoretical nature, the research outlined in thisproposal may prove useful in many areas of applied science and technology.