Electronic Journal of Probability

Evolution systems of measures and semigroup properties on evolving manifolds

Li-Juan Cheng and Anton Thalmaier

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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.

Article information

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

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

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Zentralblatt MATH identifier

Primary: 60J60: Diffusion processes [See also 58J65] 58J65: Diffusion processes and stochastic analysis on manifolds [See also 35R60, 60H10, 60J60] 53C44: Geometric evolution equations (mean curvature flow, Ricci flow, etc.)

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

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Cheng, Li-Juan; Thalmaier, Anton. Evolution systems of measures and semigroup properties on evolving manifolds. Electron. J. Probab. 23 (2018), paper no. 20, 27 pp. doi:10.1214/18-EJP147. https://projecteuclid.org/euclid.ejp/1519722149

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