New Laplacian comparison theorem and its applications to diffusion processes on Riemannian manifolds
New Laplacian comparison theorem and its applications to diffusion processes on Riemannian manifolds
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DOI：
10.1112/blms.12568
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发表时间：
2020-01
影响因子：
0.9
通讯作者：
K. Kuwae;Xiang-Dong Li
中科院分区：
文献类型：
--
作者：
K. Kuwae;Xiang-Dong Li
设L=Δ -⟨∇φ,∇·⟩$L=\Delta -\langle \nabla \phi , \nabla \cdot \rangle$是一个对称扩散算子，其不变量μ(dx)=e−φ (x)m(dx) $\mu (\mathrm{d}x)=e^{-\phi (x)}\mathfrak {m} (\mathrm{d}x)$在完全非紧光滑黎曼流形（m,g） $(M,g)$上，其体积元m=volg $\mathfrak {m} =\text{\rm vol}_g$，并且φ∈C2(m) $\phi \in C^2(M)$是一个势函数。在具有CD(K,m) ${\rm CD}(K, m)$‐条件m≤1 $m\leqslant 1$和连续函数K $K$的加权完全黎曼流形上证明了一个拉普拉斯比较定理。作为结果，我们给出了m $m$‐Bakry -Émery Ricci张量对于m≥1 $m\leqslant 1$的最优条件，使得（加权）Myers定理、Bishop-Gromov体积比较定理、L $L$‐扩散过程的随机完备性和Feller性质在加权完全黎曼流形上成立。其中一些结果在m $m$‐Bakry -Émery对于m或n $m\geqslant n$的Ricci曲率进行了很好的研究(Li， J. Math。纯苹果。(9) 84 (2005), 1295-1361；洛特，评论。数学。Helv. 78 (2003), 865-883；钱其杰。数学。48 (1987),235-242；魏和威利，J.微分地球，83(2009),377-405)或m=1 $m=1$ （Wylie，译）。美国人。数学。Soc. 369 (2017), 6661-6681；Wylie and D. yerosshkin，预印本)。当m<1 $m<1$时，我们的结果在文献中是新的。
Let L=Δ−⟨∇ϕ,∇·⟩$L=\Delta -\langle \nabla \phi , \nabla \cdot \rangle$ be a symmetric diffusion operator with an invariant measure μ(dx)=e−ϕ(x)m(dx)$\mu (\mathrm{d}x)=e^{-\phi (x)}\mathfrak {m} (\mathrm{d}x)$ on a complete non‐compact smooth Riemannian manifold (M,g)$(M,g)$ with its volume element m=volg$\mathfrak {m} =\text{\rm vol}_g$ , and ϕ∈C2(M)$\phi \in C^2(M)$ a potential function. In this paper, we prove a Laplacian comparison theorem on weighted complete Riemannian manifolds with CD(K,m)${\rm CD}(K, m)$ ‐condition for m⩽1$m\leqslant 1$ and a continuous function K$K$ . As consequences, we give the optimal conditions on m$m$ ‐Bakry–Émery Ricci tensor for m⩽1$m\leqslant 1$ such that the (weighted) Myers' theorem, Bishop–Gromov volume comparison theorem, stochastic completeness and Feller property of L$L$ ‐diffusion processes hold on weighted complete Riemannian manifolds. Some of these results were well studied for m$m$ ‐Bakry–Émery Ricci curvature for m⩾n$m\geqslant n$ (Li, J. Math. Pures Appl. (9) 84 (2005), 1295–1361; Lott, Comment. Math. Helv. 78 (2003), 865–883; Qian, Q. J. Math. 48 (1987), 235–242; Wei and Wylie, J. Differential Geom. 83 (2009), 377–405) or m=1$m=1$ (Wylie, Trans. Amer. Math. Soc. 369 (2017), 6661–6681; Wylie and D. Yeroshkin, Preprint). When m<1$m<1$ , our results are new in the literature.