# 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.

## Key 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.