Reference : Evolution systems of measures and semigroup properties on evolving manifolds
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Physical, chemical, mathematical & earth Sciences : Mathematics
http://hdl.handle.net/10993/31923
Evolution systems of measures and semigroup properties on evolving manifolds
English
Cheng, Li Juan mailto [University of Luxembourg > Faculty of Science, Technology and Communication (FSTC) > Mathematics Research Unit >]
Thalmaier, Anton mailto [University of Luxembourg > Faculty of Science, Technology and Communication (FSTC) > Mathematics Research Unit >]
16-Aug-2017
22
No
[en] Evolution system of measures ; geometric flow ; inhomogeneous diffusion ; semigroup ; supercontractivity ; hypercontractivity ; ultraboundedness
[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.
Researchers ; Professionals
http://hdl.handle.net/10993/31923
http://arxiv.org/abs/1708.04951
https://arxiv.org/abs/1708.04951
FnR ; FNR7628746 > Anton Thalmaier > GEOMREV > Geometry Of Random Evolutions > 01/03/2015 > 28/02/2018 > 2014

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