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 $P_tf$ vanishes at infinity for all $t > 0$ whenever $f$ is so. Let $x_0 \in M$ and let $\kappa(r)^2 \geq -\inf\{Ric(x): \rho(x, x_0) \leq r\}$ ($\rho$ is the Riemannian distance). We prove that $M$ is stochastically complete and has the $C_0$-diffusion property if $\int^\infty_c \kappa(r)^{-1} dr = \infty$ by studying the radial part of the Riemannian Brownian motion on $M$.