# Derivative and divergence formulae for diffusion semigroups

## TL;DR

This paper derives probabilistic formulas for diffusion semigroups on manifolds, enabling analysis of heat kernel derivatives and applications like shift-Harnack inequalities, using martingale methods without involving derivatives of functions.

## Contribution

It introduces new probabilistic formulae for diffusion semigroups that do not involve derivatives of functions, applicable to both symmetric and non-symmetric generators.

## Key findings

- Derived probabilistic formulas for $P_t(V(f))$ using martingale arguments
- Formulas relate to derivatives of the heat kernel in the forward variable
- Applications include deriving shift-Harnack inequalities

## Abstract

For a semigroup $P_t$ generated by an elliptic operator on a smooth manifold $M$, we use straightforward martingale arguments to derive probabilistic formulae for $P_t(V(f))$, not involving derivatives of $f$, where $V$ is a vector field on $M$. For non-symmetric generators, such formulae correspond to the derivative of the heat kernel in the forward variable. As an application, these formulae can be used to derive various shift-Harnack inequalities.