Open Access
July, 1989 Heat Semigroup on a Complete Riemannian Manifold
Pei Hsu
Ann. Probab. 17(3): 1248-1254 (July, 1989). DOI: 10.1214/aop/1176991267


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 $M$.


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Pei Hsu. "Heat Semigroup on a Complete Riemannian Manifold." Ann. Probab. 17 (3) 1248 - 1254, July, 1989.


Published: July, 1989
First available in Project Euclid: 19 April 2007

zbMATH: 0694.58043
MathSciNet: MR1009455
Digital Object Identifier: 10.1214/aop/1176991267

Primary: 58J32

Keywords: $C_0$-diffusion , comparison theorems , Ricci curvature , Riemannian Brownian motion , Riemannian manifold , stochastic completeness

Rights: Copyright © 1989 Institute of Mathematical Statistics

Vol.17 • No. 3 • July, 1989
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