This proposal is primarily concerned with three topics. The first is to study the smoothness properties of densities for laws of solutions to hypoelliptic stochastic differential equations driven by fractional Brownian motion. The second is to continue to analyze certain finite dimensional geometric approximations to Wiener measure on a Riemannian manifold. The eventual goal is to include the super-symmetric setting which will likely entail analysis in non-commutative probability spaces. The P.I. expects this aspect of the project will make contact with the theory of random matrices. The third problem is devoted to proving parabolic regularity results for heat equations on certain infinite dimensional Lie groups including loop groups. In one way or another each of these problems may be recast (at least heuristically) as problem involving path integrals. Many of the projects proposed here are loosely motivated by the fundamental problem of constructing quantize Yang-Mills fields. See the Clay Mathematics Institute problem pertaining to quantized Yang - Mills fields for a description of this problem and its importance to the so called standard model in physics.
Fractional Brownian motion (fBm) was implicitly introduced by A. N. Kolmogorov in 1940. (Kolmogorov seemed to have in mind possible applications to the theory of turbulence.) A later paper by Mandelbrot and Van Ness (1968) describes a number of possible applications for fBm including using it for economic and hydrology models. The recent literature includes applications of fBm in the modeling of neural networks, the pricing of financial instruments, and to understanding the clustering of galaxies. One of the main projects of this proposal is to study the statistical behavior of these types of dynamical systems which are driven by fractional Brownian motion. The P.I. will also focus on two other projects during the period of this research grant. One of these projects is a continuation of the P.I.'s program to better understand the Feynman "path integral" techniques which are used in the quantum mechanical description of atomic and sub-atomic particles. Although highly studied, the mathematical footing of Feynman's path integrals is still tenuous at best. The third project is to generalize the results of the first two projects to "infinite dimensions." It turns out that the seemingly very abstract generalization to infinite dimensions is precisely what is needed in order to understand the quantum mechanical description of the fundamental forces in nature. (This topic goes under the heading of quantum field theory in the particle physics literature.) It is a long standing major challenge to make mathematically rigorous sense of interacting quantum field theories. This proposal has a significant graduate training component as a number of the problems will be tackled by the P.I.'s students.
