Robin Heat Semigroup and HWI Inequality on Manifolds with Boundary

TL;DR

This paper develops a probabilistic approach to analyze the Robin heat semigroup on manifolds with boundary, establishing the HWI inequality and gradient estimates for noncompact manifolds with boundary.

Contribution

It introduces a probabilistic formula for the Robin heat semigroup using reflecting diffusion processes and derives the HWI inequality on manifolds with boundary, including nonconvex cases.

Findings

Probabilistic formula for Robin heat semigroup using reflecting diffusion.

HWI inequality established on manifolds with boundary.

Gradient estimates and Bismut's derivative formula for noncompact manifolds.

Abstract

Let $M$ be a complete connected Riemannian manifold with boundary $\pp M$, $Q$ a bounded continuous function on $\pp M$, and $L= \DD+Z$ for a $C^1$-vector field $Z$ on $M$. By using the reflecting diffusion process generated by $L$ and its local time on the boundary, a probabilistic formula is presented for the semigroup generated by $L$ on $M$ with Robin boundary condition $N,\nn f+Qf=0,$ where $N$ is the inward unit normal vector field of $\pp M$. As an application, the HWI inequality is established on manifolds with (nonconvex) boundary. In order to study this semigroup, Hsu's gradient estimate and the corresponding Bismut's derivative formula are established on a class of noncompact manifolds with boundary.