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Diffusion semigroup on manifolds with time-dependent metrics

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Veröffentlicht/Copyright: 18. August 2016
Forum Mathematicum
Aus der Zeitschrift Forum Mathematicum Band 29 Heft 4

Abstract

Let Lt:=Δt+Zt, t∈[0,Tc) on a differential manifold equipped with a complete geometric flow (gt)t∈[0,Tc), where Δt is the Laplacian operator induced by the metric gt and (Zt)t∈[0,Tc) is a family of C1,∞-vector fields. In this article, we present a number of equivalent inequalities for the lower bound curvature condition, which include gradient inequalities, transportation-cost inequalities, Harnack inequalities and other functional inequalities for the semigroup associated with diffusion processes generated by Lt. To this end, we establish derivative formulae for the associated semigroup and construct coupling processes for these diffusion processes by parallel displacement and reflection.

Keywords: geometric flow; curvature; coupling; transportation-cost inequality, Harnack inequality; gradient inequality
MSC 2010: 60J60; 58J65; 53C44

Communicated by Ichiro Shigekawa

Funding source: National Natural Science Foundation of China

Award Identifier / Grant number: A011002

Funding source: Natural Science Foundation of Zhejiang Province

Award Identifier / Grant number: LQ16A010009

Funding source: Zhejiang University of Technology

Award Identifier / Grant number: 2014XZ011

Funding statement: The author was supported in part by the National Natural Science Foundation of China (grant no. A011002), the Natural Science Foundation of Zhejiang Province (grant no. LQ16A010009) and the Natural Science Foundation of Zhejiang University of Technology (grant no. 2014XZ011).

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Received: 2015-3-18
Revised: 2016-4-8
Published Online: 2016-8-18
Published in Print: 2017-7-1

© 2017 Walter de Gruyter GmbH, Berlin/Boston

Artikel in diesem Heft

  1. Frontmatter
  2. Hurewicz fibrations, almost submetries and critical points of smooth maps
  3. Lower bounds for regular genus and gem-complexity of PL 4-manifolds
  4. Diffusion semigroup on manifolds with time-dependent metrics
  5. Blow-up algebras, determinantal ideals, and Dedekind–Mertens-like formulas
  6. A computational approach to Milnor fiber cohomology
  7. Metrical universality for groups
  8. The second and third moment of L(1/2,χ) in the hyperelliptic ensemble
  9. Slender domains and compact domains
  10. Topological 2-generation of automorphism groups of countable ultrahomogeneous graphs
  11. A note on local Hardy spaces
  12. On the value group of a model of Peano Arithmetic
  13. Extremal values of the (fractional) Weinstein functional on the hyperbolic space
  14. Maximal subsemigroups of some semigroups of order-preserving mappings on a countably infinite set
  15. Unitals in shift planes of odd order
https://doi.org/10.1515/forum-2015-0049
Schlagwörter für diesen Artikel
geometric flow; curvature; coupling; transportation-cost inequality, Harnack inequality; gradient inequality

Artikel in diesem Heft

  1. Frontmatter
  2. Hurewicz fibrations, almost submetries and critical points of smooth maps
  3. Lower bounds for regular genus and gem-complexity of PL 4-manifolds
  4. Diffusion semigroup on manifolds with time-dependent metrics
  5. Blow-up algebras, determinantal ideals, and Dedekind–Mertens-like formulas
  6. A computational approach to Milnor fiber cohomology
  7. Metrical universality for groups
  8. The second and third moment of L(1/2,χ) in the hyperelliptic ensemble
  9. Slender domains and compact domains
  10. Topological 2-generation of automorphism groups of countable ultrahomogeneous graphs
  11. A note on local Hardy spaces
  12. On the value group of a model of Peano Arithmetic
  13. Extremal values of the (fractional) Weinstein functional on the hyperbolic space
  14. Maximal subsemigroups of some semigroups of order-preserving mappings on a countably infinite set
  15. Unitals in shift planes of odd order
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