Heat Semigroup on a Complete Riemannian Manifold

Abstract

Let $M$ be a complete Riemannian manifold and $p(t,x,y)$ the minimal heat kernel on $M$. Let $P_t$ be the associated semigroup. We say that $M$ is stochastically complete if ${\int }_{M}p(t,x,y)dy=1$ for all $t>0,x\in M$; we say that $M$ has the $C_0$-diffusion property (or the Feller property) if $P_{t}f$ vanishes at infinity for all $t>0$ whenever $f$ is so. Let $x_0\in M$ and let $\kappa (r{)}^2\ge -inf\{Ric(x):\rho (x,x_0)\le r\}$ ($\rho$ is the Riemannian distance). We prove that $M$ is stochastically complete and has the $C_0$-diffusion property if ${\int }_c^{\mathrm{\infty }}\kappa (r{)}^{-1}dr=\mathrm{\infty }$ by studying the radial part of the Riemannian Brownian motion on $M$.