Generalized Feynman-Kac Formula and Associated Heat Kernel
Abstract
Let M be a smooth closed (compact without boundary) Riemannian manifold of dimension n and P a q-dimensional smooth submanifold of M. U will denote the tubular neighborhood of P in M. Let E be a smooth vector bundle over M. Here we will obtain a vector bundle Generalized Feynman-Kac formula associated to U and the vector bundle differential operator L consisteing of half the generalized Laplacian, a vector field X on M and a potential term V on M. From this formula, we shall deduce the usual Feynman-Kac formula as well as a stochastic representation of the Generalized Elworthy-Truman Heat Kernel formula, and ultimately the heat kernel formula. The Feynman-Kac expression can be expanded and from this expansion we shall deduce both the generalized heat trace and heat content expansions. The Generalized Feynman-Kac Formula is thus at the center of several other previously known results.
Cite
@article{arxiv.2306.04740,
title = {Generalized Feynman-Kac Formula and Associated Heat Kernel},
author = {Martin N. Ndumu},
journal= {arXiv preprint arXiv:2306.04740},
year = {2025}
}
Comments
This article has 381 pages and 1 figure (at the beginning of the article). It was written a long time ago but given its volume due to lengthy computations, it took a long time to correct some of the computational errors