Formulae for the derivatives of heat semigroups

TL;DR

This paper derives differentiation formulas for heat semigroups using martingale methods, applicable on Euclidean spaces and manifolds, including for heat equations on differential forms and second order derivatives.

Contribution

It introduces a martingale-based approach to obtain derivative formulas for heat semigroups on manifolds and differential forms, extending previous methods.

Findings

Derived a formula for the first derivative of heat semigroups on R^n and manifolds.

Extended the differentiation formulas to heat equations on differential forms.

Established second order differentiation formulas for heat semigroups.

Abstract

We use a basic martingale method to show a differentiation formula for the derivatives $$d(P_tf)(x_0)(v_0)={1\over t} E f(x_t) \int_0^t \langle Y(x_s)(v_s),dB_t\rangle_{R^m}.$$ These are proved first on $R^n$, then on manifolds. Afterwards for solutions of heat equations on differential forms, and a second order formula.