Bismut's gradient formula for vector bundles

TL;DR

This paper extends Bismut's gradient formula to vector bundles over compact Riemannian manifolds, enabling probabilistic representations of higher order derivatives of heat equation solutions.

Contribution

It introduces a general Bismut's formula for vector bundles, broadening the scope of probabilistic gradient representations beyond scalar cases.

Findings

Derived a new Bismut's formula for vector bundles

Enabled probabilistic representation of higher order derivatives

Applicable to solutions of the heat equation on manifolds

Abstract

We prove a general Bismut's formula for the gradient of a class of smooth Wiener functionals over vector bundles of a compact Riemannian manifold. This general formula can be used repeatedly for obtaining probabilistic representation of higher order covariant derivatives of solutions of the heat equation similar to the classical Bismut's representation for the covariant gradient of the heat kernel.