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2018 Evolution systems of measures and semigroup properties on evolving manifolds
Li-Juan Cheng, Anton Thalmaier
Electron. J. Probab. 23: 1-27 (2018). DOI: 10.1214/18-EJP147


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.


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Li-Juan Cheng. Anton Thalmaier. "Evolution systems of measures and semigroup properties on evolving manifolds." Electron. J. Probab. 23 1 - 27, 2018.


Received: 16 August 2017; Accepted: 2 February 2018; Published: 2018
First available in Project Euclid: 27 February 2018

zbMATH: 1390.60287
MathSciNet: MR3771757
Digital Object Identifier: 10.1214/18-EJP147

Primary: 53C44 , 58J65 , 60J60

Keywords: evolution system of measures , geometric flow , hypercontractivity , inhomogeneous diffusion , semigroup , supercontractivity , ultraboundedness


Vol.23 • 2018
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