Robin Heat Semigroup and HWI Inequality on Manifolds with Boundary

Details Robin Heat Semigroup and HWI Inequality on Manifolds with Boundary Robin Heat Semigroup and HWI Inequality on Manifolds with Boundary

2009-08-20

arxiv.org/abs/0908.2892v1

Mathematics

Probability

22 pages

Scientific paper

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.

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