Derivatives of Feynman-Kac Semigroups

TL;DR

This paper establishes Bismut-type formulas for the derivatives of Feynman-Kac semigroups on Riemannian manifolds, providing local estimates and bounds, with applications to stationary solutions, all based on geometric assumptions.

Contribution

It introduces new Bismut-type formulas for derivatives of Feynman-Kac semigroups on manifolds under purely geometric conditions.

Findings

Derived formulas for first and second derivatives.

Provided bounds on logarithmic derivatives of kernels.

Extended analysis to stationary solutions.

Abstract

We prove Bismut-type formulae for the first and second derivatives of a Feynman-Kac semigroup on a complete Riemannian manifold. We derive local estimates and give bounds on the logarithmic derivatives of the integral kernel. Stationary solutions are also considered. The arguments are based on local martingales, although the assumptions are purely geometric.