CAREER: Heat kernel measures in infinite dimensions

University Of Virginia Main Campus, Charlottesville VA

Investigators

Abstract

This research project is unified as a study of smoothness properties of elliptic and hypoelliptic heat kernel measures in infinite dimensions. The focal points of this proposal - smoothness, quasi-invariance, and Taylor isomorphisms - are fundamental principles in the study of measures in infinite dimensions. Although these concepts have their own intrinsic interest, one may motivate their study by physical applications. For example, in quantized classical mechanics, it is standard practice to perform integration over certain infinite-dimensional spaces with respect to "infinite-dimensional Lebesgue measure." Of course, this measure does not exist, and the appropriate replacement is Wiener measure or, for an infinite-dimensional curved space, heat kernel measure. As the integral computations physicists perform routinely involve changes of variable and differentiation, quasi-invariance and more general smoothness results for heat kernel measures are critical in giving a mathematical foundation to these formal computations. Taylor isomorphism theorems are also significant to physical applications as they are related to the important problem of quantizing Yang-Mills fields which form a key part of the "standard model" of particle physics. The study of hypoelliptic heat kernel measures are important in the study of sub-Riemannian geometries that appear in physical models. This is an integrated project of research and training in probability. The research program focuses on the study of so-called ?heat kernel measures? in infinite dimensions. Consider a curved surface (like the Earth) to which one applies a unit of heat at a fixed point and then steps away and allows the heat to propagate. The heat kernel measure describes the evolution of the proportion of heat in subsets of the surface at some later time. As the manner in which heat diffuses relies heavily on the geometric and topological properties of the relevant space, the heat kernel measure reflects these properties and analysis of this measure reveals much about the space on which it lives. Heat kernel measures are the natural generalization to curved spaces of Gaussian measures (higher dimensional analogues of the normal distribution) which have for many decades been recognized as the appropriate measures to conduct infinite-dimensional analysis. Infinite-dimensional spaces often show up in physical models; in particular, infinite-dimensional groups appear in quantum field theory and quantum mechanics. Thus, this research program contributes to solidifying a rigorous mathematical framework for the study of quantum physics. Many of the applications have finite-dimensional models which are accessible to undergraduates or beginning graduate students. Thus, this project has a significant training component, including involving undergraduates in research on related problems in finite dimensions. The training program also includes mentoring of graduate students in an introduction to math outreach and promoting the retention and visibility of women researchers in probability.