The differentiation of hypoelliptic diffusion semigroups

TL;DR

This paper develops derivative formulas for hypoelliptic heat semigroups, extending elliptic results, and introduces integration by parts techniques using local martingales to handle boundary conditions and diffusion lifetime.

Contribution

It provides new derivative formulas for hypoelliptic semigroups based on Malliavin calculus, extending previous elliptic case results and addressing boundary and finite lifetime issues.

Findings

Derived derivative formulas for hypoelliptic heat semigroups.

Developed integration by parts formulas at the level of local martingales.

Extended methods to nonlinear harmonic mappings.

Abstract

Basic derivative formulas are presented for hypoelliptic heat semigroups and harmonic functions extending earlier work in the elliptic case. Emphasis is placed on developing integration by parts formulas at the level of local martingales. Combined with the optional sampling theorem, this turns out to be an efficient way of dealing with boundary conditions, as well as with finite lifetime of the underlying diffusion. Our formulas require hypoellipticity of the diffusion in the sense of Malliavin calculus (integrability of the inverse Malliavin covariance) and are formulated in terms of the derivative flow, the Malliavin covariance and its inverse. Finally some extensions to the nonlinear setting of harmonic mappings are discussed.