KMS Of Academy of mathematics and systems sciences, CAS
| New Laplacian comparison theorem and its applications to diffusion processes on Riemannian manifolds | |
Kuwae, Kazuhiro1; Li, Xiang-Dong2,3
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| 2022-03-11 | |
| Source Publication | BULLETIN OF THE LONDON MATHEMATICAL SOCIETY
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| ISSN | 0024-6093 |
| Pages | 24 |
| Abstract | 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 |
| Indexed By | SCI |
| Language | 英语 |
| Funding Project | 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 Research Area | Mathematics |
| WOS Subject | Mathematics |
| WOS ID | WOS:000767468200001 |
| Publisher | WILEY |
| Citation statistics | |
| Document Type | 期刊论文 |
| Identifier | http://ir.amss.ac.cn/handle/2S8OKBNM/60128 |
| Collection | 应用数学研究所 |
| Corresponding Author | Kuwae, Kazuhiro |
| Affiliation | 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 |
| Recommended Citation 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. |
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