New Laplacian comparison theorem and its applications to diffusion processes on Riemannian manifolds

TL;DR

This paper establishes a new Laplacian comparison theorem for weighted Riemannian manifolds under the ${ m CD}(K, m)$-condition with $m eq n$, leading to optimal geometric and stochastic properties for diffusion processes.

Contribution

It introduces a novel Laplacian comparison theorem for weighted manifolds with ${ m CD}(K, m)$-condition for $m eq n$, extending previous results and deriving new geometric and probabilistic implications.

Findings

Proves a Laplacian comparison theorem under ${ m CD}(K, m)$ for $m eq n$.

Derives optimal conditions for Myers' theorem, volume comparison, and splitting theorems.

Establishes stochastic completeness and Feller property for $L$-diffusions under new conditions.

Abstract

Let $L=\Delta-\nabla\phi\cdot \nabla$ be a symmetric diffusion operator with an invariant measure $\mu({\rm} d x)=e^{-\phi(x)}{\mathfrak m}({\rm d} x)$ on a complete non-compact smooth Riemannian manifold $(M,g)$ with its volume element ${\mathfrak m}={\rm vol}_g$, and $\phi\in C^2(M)$ a potential function. In this paper, we prove a Laplacian comparison theorem on weighted complete Riemannian manifolds with ${\rm CD}(K, m)$-condition for $m\leq 1$ and a continuous function $K$. As consequences, we give the optimal conditions on $m$-Bakry-\'Emery Ricci tensor for $m\leq1$ such that the (weighted) Myers' theorem, Bishop-Gromov volume comparison theorem, Ambrose-Myers' theorem, and the Cheeger-Gromoll type splitting theorem, stochastic completeness and Feller property of $L$-diffusion processes hold on weighted complete Riemannian manifolds. Some of these results were well-studied for $m$-Bakry-\'Emery Ricci curvature for $m\geq n$ (\!\!\cite{Lot,Qian,XDLi05, WeiWylie}) or $m=1$ (\!\!\cite{Wylie:WarpedSplitting, WylieYeroshkin}). When $m<1$, our results are new in the literature.