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

doi:10.1112/blms.12568

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	New Laplacian comparison theorem and its applications to diffusion processes on Riemannian manifolds
	Kuwae, Kazuhiro 1; Li, Xiang-Dong2,3
	2022-03-11
发表期刊	BULLETIN OF THE LONDON MATHEMATICAL SOCIETY
ISSN	0024-6093
页码	24
摘要	Let L=Delta-⟨ backward difference phi, backward difference center dot⟩$L=\Delta -\langle \nabla \phi , \nabla \cdot \rangle$ be a symmetric diffusion operator with an invariant measure mu(dx)=e-phi(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 phi is an element of 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-emery 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-emery 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.
DOI	10.1112/blms.12568
收录类别	SCI
语种	英语
资助项目	JSPS[17H02846] ; Central Research Institute of Fukuoka University[185001] ; National Key R&D Program of China[2020YFA0712700] ; NSFC[12171458] ; NSFC[11771430] ; NSFC[11688101] ; Key Laboratory RCSDS, CAS[2008DP173182]
WOS研究方向	Mathematics
WOS类目	Mathematics
WOS记录号	WOS:000767468200001
出版者	WILEY
引用统计
文献类型	期刊论文
条目标识符	http://ir.amss.ac.cn/handle/2S8OKBNM/60128
专题	应用数学研究所
通讯作者	Kuwae, Kazuhiro
作者单位	1.Fukuoka Univ, Dept Appl Math, Fukuoka 8140180, Japan 2.Chinese Acad Sci, Acad Math & Syst Sci, Beijing, Peoples R China 3.Univ Chinese Acad Sci, Sch Math Sci, Beijing, Peoples R China
推荐引用方式 GB/T 7714	Kuwae, Kazuhiro,Li, Xiang-Dong. New Laplacian comparison theorem and its applications to diffusion processes on Riemannian manifolds[J]. BULLETIN OF THE LONDON MATHEMATICAL SOCIETY,2022:24.
APA	Kuwae, Kazuhiro,&Li, Xiang-Dong.(2022).New Laplacian comparison theorem and its applications to diffusion processes on Riemannian manifolds.BULLETIN OF THE LONDON MATHEMATICAL SOCIETY,24.
MLA	Kuwae, Kazuhiro,et al."New Laplacian comparison theorem and its applications to diffusion processes on Riemannian manifolds".BULLETIN OF THE LONDON MATHEMATICAL SOCIETY (2022):24.