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

Abstract

Let $L=\Delta -\nabla \varphi \cdot \nabla$ be a symmetric diffusion operator with an invariant measure $\mu (dx)=e^{-\varphi (x)}\mathfrak{m}(\mathrm{d}x)$ on a complete non-compact smooth Riemannian manifold $(M,g)$ with its volume element $\mathfrak{m}={\mathrm{vol}}_g$, and $\varphi \in C^2(M)$ a potential function. In this paper, we prove a Laplacian comparison theorem on weighted complete Riemannian manifolds with $\mathrm{CD}(K,m)$-condition for $m\le 1$ and a continuous function $K$. As consequences, we give the optimal conditions on $m$-Bakry-Émery Ricci tensor for $m\le 1$ 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-Émery Ricci curvature for $m\ge n$ (\!\!\cite{Lot,Qian,XDLi05, WeiWylie}) or $m=1$ (\!\!\cite{Wylie:WarpedSplitting, WylieYeroshkin}). When $m<1$, our results are new in the literature.