Generalized Feynman-Kac Formula and Associated Heat Kernel

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.

💡 Research Summary

The paper develops a comprehensive stochastic representation for the heat semigroup generated by a very general second‑order elliptic operator acting on sections of a smooth vector bundle over a closed Riemannian manifold. Let (M) be a compact, boundary‑free Riemannian manifold of dimension (n) and (P\subset M) a smooth submanifold of dimension (q). A tubular neighbourhood (U) of (P) is fixed. On a vector bundle (E\to M) the authors consider the operator

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