A ug 2 00 9 Robin Heat Semigroup and HWI Inequality on Manifolds with Boundary ∗

Let M be a complete connected Riemannian manifold with boundary ∂M , Q a bounded continuous function on ∂M , and L = ∆+Z for a C1-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,∇f〉+ Qf = 0, where N is the inward unit normal vector field of ∂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. AMS subject Classification: 60J60, 58G32.