Heat Kernel Analysis

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This research is devoted to questions involving heat equations in both finite and infinite dimensional contexts. The following issues are to be addressed. 1. The existence of differential type inequalities related to hypoelliptic heat equations in finite dimensions, including the existence of logarithmic Sobolev and Poincare (or mass gap) type inequalities. 2. The existence of differential inequalities related to heat kernel measure and Wiener measures on path and loop spaces of Riemannian manifolds 3. The relationship between heat kernel measures and pinned Wiener measures on Loop spaces of compact Riemannian manifolds. 4. Extensions of Fock space like representations from finite dimensional groups to infinite dimensional loop groups. 5. The generalization of certain finite dimensional approximations to Wiener measures to include the "super symmetric quantum mechanics" setting used by physicists. Most of the questions to be studied in this research are motivated by, or are outgrowths of, questions coming from studying a number of fields, namely the spread of heat in curved metal plates, quantum mechanics, quantum field theories, and certain aspects of probability theory. These seemingly disparate fields turn out to share a common mathematical description, namely "parabolic partial differential equations" or equivalently the probabilistic theory of "Brownian motion." In most interesting (physically relevant) cases it is seldom possible to find explicit solutions to the complicated partial differential equations to be studied by the P.I. Nevertheless, it is often possible to discover interesting and relevant properties of the solutions. A typical phenomenon of parabolic partial differential equations is the fact that their solutions tend towards steady state values after waiting a sufficiently long time. For example, if a metal plate is heated in a non-uniform way and then left alone, the heat in the plate will redistribute itself over time so that the temperature reaches a constant equilibrium value throughout the plate. A fundamental problem, related to "Poincare" and "logarithmic Sobolev" inequalities, is the questions of how quickly do the systems described by the parabolic partial differential equations to be studied in this research converge to their equilibrium value. This same rate of convergence question has another interpretation in the context of quantum field theories describing relativistic (i.e. moving near the speed of light) elementary particles. For these theories the existence of a Poincare inequality has the desirable interpretation that the mathematical theory does not predict an unreasonable plethora of elementary particles. The particle interpretation of parabolic and related quantum mechanical Shrodinger equations will be another point of study. The particle interpretation goes under the title of Fock space representations which were mentioned in the first paragraph.