Electronic Journal of Probability

Evolution systems of measures and semigroup properties on evolving manifolds

Li-Juan Cheng and Anton Thalmaier

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Abstract

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1519722149

Digital Object Identifier
doi:10.1214/18-EJP147

Mathematical Reviews number (MathSciNet)
MR3771757

Zentralblatt MATH identifier
1390.60287

Subjects
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.)

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

Rights
Creative Commons Attribution 4.0 International License.

Citation

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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