Second Order Bismut formulae and applications to Neumann semigroups on manifolds

TL;DR

This paper derives second order Bismut formulae for Neumann semigroups on manifolds with boundary, providing new estimates and inequalities that connect curvature, entropy, and Fisher information.

Contribution

The work introduces second order Bismut formulae for Neumann semigroups on manifolds with boundary, extending previous results to include boundary effects and new functional inequalities.

Findings

Derived Bismut formulae for $LP_t f$ and Hessian of $P_t f$

Established estimates under curvature conditions

Proved a new log-Sobolev inequality linking entropy, discrepancy, and Fisher information

Abstract

Let $M$ be a complete connected Riemannian manifold with boundary $\partial M$, and let $P_t$ be the Neumann semigroup generated by $\frac{ 1}{ 2} L$ where $L=\Delta+Z$ for a $C^1$-vector field $Z$ on $M$. We establish Bismut type formulae for $LP_t f$ and ${\rm Hess}_{P_tf}$ and present estimates of these quantities under suitable curvature conditions. In case when $P_t$ is symmetric in $L^2(\mu)$ for some probability measure $\mu$, a new type of log-Sobolev inequality is established which links the relative entropy $H$, the Stein discrepancy $S$, and relative Fisher information $I$, generalizing the authors' recent work in the case without boundary.