Evolution systems of measures and semigroup properties on evolving manifolds

TL;DR

This paper investigates the properties of diffusion processes on evolving manifolds, establishing conditions for their non-explosion, existence, uniqueness, and functional inequalities of associated semigroups.

Contribution

It provides new criteria for non-explosion and existence of measures, and characterizes semigroup properties on evolving manifolds using coupling and Sobolev inequalities.

Findings

Conditions for non-explosion of diffusion processes

Existence and uniqueness of evolution systems of measures

Characterization of supercontractivity and hypercontractivity

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.