Article (Scientific journals)
Evolution systems of measures and semigroup properties on evolving manifolds
Cheng, Li Juan; Thalmaier, Anton
2018In Electronic Journal of Probability, 23 (20), p. 1-27
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Keywords :
Evolution system of measures; geometric flow; inhomogeneous diffusion; semigroup; supercontractivity; hypercontractivity; ultraboundedness
Abstract :
[en] An evolving Riemannian manifold (M,g_t)_{t\in I} consists of a smooth d-dimensional manifold M, equipped with a geometric flow g_t of complete Riemannian metrics, parametrized by I=(-\infty,T). Given an additional C^{1,1} family of vector fields (Z_t)_{t\in I} on M. We study the family of operators L_t=\Delta_t +Z_t where \Delta_t denotes the Laplacian with respect to the metric g_t. We first give sufficient conditions, in terms of space-time Lyapunov functions, for non-explosion of the diffusion generated by L_t, and for existence of evolution systems of probability measures associated to it. Coupling methods are used to establish uniqueness of the evolution systems under suitable curvature conditions. Adopting such a unique system of probability measures as reference measures, we characterize supercontractivity, hypercontractivity and ultraboundedness of the corresponding time-inhomogeneous semigroup. To this end, gradient estimates and a family of (super-)logarithmic Sobolev inequalities are established.
Disciplines :
Mathematics
Author, co-author :
Cheng, Li Juan ;  University of Luxembourg > Faculty of Science, Technology and Communication (FSTC) > Mathematics Research Unit
Thalmaier, Anton ;  University of Luxembourg > Faculty of Science, Technology and Communication (FSTC) > Mathematics Research Unit
External co-authors :
no
Language :
English
Title :
Evolution systems of measures and semigroup properties on evolving manifolds
Publication date :
27 February 2018
Journal title :
Electronic Journal of Probability
ISSN :
1083-6489
Publisher :
Institute of Mathematical Statistics, Beachwood, United States - Ohio
Volume :
23
Issue :
20
Pages :
1-27
Peer reviewed :
Peer Reviewed verified by ORBi
FnR Project :
FNR7628746 - Geometry Of Random Evolutions, 2014 (01/03/2015-28/02/2018) - Anton Thalmaier
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