论文标题
新的laplacian比较定理及其在riemannian流形的扩散过程中的应用
New Laplacian comparison theorem and its applications to diffusion processes on Riemannian manifolds
论文作者
论文摘要
令$ l =δ-\ nablaϕ \ cdot \ nabla $为对称扩散操作员，具有不变的度量$μ（{\ rm} d x）= e^{ - ϕ（x）} {\ mathfrak m}（{\ rm d} x）$ complock $ riem $ rieman $ rieman clist $ rieman clist uliem clist ulian uliean uliem clist clist clist $ riem cold collate $ riem clist clield $ riem clist collate $（ $ {\ mathfrak m} = {\ rm vol} _g $，而在c^2（m）$中$ ϕ \ a潜在功能。在本文中，我们证明了具有$ {\ rm cd}（k，m）$的加权完整riemannian歧管的拉普拉斯比较定理 - $ m \ leq 1 $的条件和连续功能$ k $。 As consequences, we give the optimal conditions on $m$-Bakry-Émery 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里曼尼亚人的歧管。其中一些结果以$ m $ m $ bakry-émeryricci曲率进行了充分研究，$ m \ geq n $（\！\！\！当$ m <1 $时，我们的结果在文献中是新的。
Let $L=Δ-\nablaϕ\cdot \nabla$ be a symmetric diffusion operator with an invariant measure $μ({\rm} d x)=e^{-ϕ(x)}{\mathfrak m}({\rm d} x)$ on a complete non-compact smooth Riemannian manifold $(M,g)$ with its volume element ${\mathfrak m}={\rm vol}_g$, and $ϕ\in C^2(M)$ 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-Émery 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-Émery 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.
