Evolution systems of measures and semigroups properties on evolving manifolds
by Li-Juan Cheng and Anton Thalmaier

An evolving Riemannian manifold (M,gt)t∈I consists of a smooth d-dimensional manifold M, equipped with a geometric flow gt of complete Riemannian metrics, parametrized by I=(-∞,T). Given an additional C1,1 family of vector fields (Zt)t∈I on M, we study the family of operators Ltt+Zt where Δt denotes the Laplacian with respect to the metric gt. We first give sufficient conditions, in terms of space-time Lyapunov functions, for non-explosion of the diffusion generated by Lt, 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.

Electron. J. Probab. 23  (2018), paper no. 20, 27 pp. 


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Li Juan Cheng
Anton Thalmaier

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