Evolution systems of measures and semigroups properties on evolving manifolds
by Li-Juan Cheng and Anton Thalmaier
Abstract
An evolving Riemannian manifold consists of a
smooth -dimensional manifold , equipped with a geometric flow
of complete Riemannian metrics, parametrized
by . Given an additional family of vector
fields on , we study the family
of operators
where denotes the Laplacian with
respect to the metric . We first give sufficient conditions,
in terms of space-time Lyapunov functions, for non-explosion of
the diffusion generated by , 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.
https://doi.org/10.1214/18-EJP147
The paper is available here:
Back to Homepage