Evolution systems of measures and semigroup properties on evolving manifolds

Abstract

An evolving Riemannian manifold (M, g t ) t∈I consists of a smooth d-dimensional manifold M, equipped with a geometric flow g t of complete Riemannian metrics, parametrized by I = (−∞, T ). Given an additional C 1,1 family of vector fields (Z t ) t∈I on M. We study the family of operators L t = ∆ t + Z t where ∆ 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.We investigate this problem from a probabilistic point of view. Let X t be the diffusion process generated by L t (called L t -diffusion) which is assumed to be non-explosive before time T (see [2,8,14] for details). As in the time-homogeneous case, we construct L t -diffusions X t via horizontal diffusions u t above X t .

Functional inequalities for the heat flow on time‐dependent metric measure spaces

“…Only recently, some of these properties and equivalences have been extended to the heat flow on time-dependent Riemannian manifolds, for example, by Cheng and Thalmaier [9], Haslhofer and Naber [12], McCann and Topping [17], and Cheng [8]. The authors of the current paper had been the first to study the heat flow on time-dependent metric measure spaces [14], to introduce the time-dependent counterpart of synthetic lower Ricci bounds, and to derive various functional inequalities equivalent to it.…”

Functional inequalities for the heat flow on time-dependent metric measure spaces

“…Since g is arbitrary, this proves the first two claims of the theorem in the case of bounded u ∈ D(E). The claim (9) for bounded u ∈ L 2 (X) follows by applying the latter estimate with s + δ in the place of s to the function P s+δ,s u as δ → 0, which lies in D(E) and from gP t,s+δ ((P s+δ,s u) 2 )dm t → gP t,s (u 2 )dm t which in turn is a consequence of the continuity of δ → P * t,s+δ g and of δ → P s+δ,s u in L 2 and the uniform boundedness of the latter in L ∞ .…”

“…Only recently, some of these properties and equivalences have been extended to the heat flow on time-dependent Riemannian manifolds, e.g. by Cheng & Thalmaier [9], Haslhofer & Naber [12],…”

Exponential contraction in Wasserstein distance on static and evolving manifolds

“…In [5], the first two authors investigated existence and uniqueness of evolution systems of measures. In particular, they found that W 1 -contraction of the distance helps to prove uniqueness properties (see [5] for details).…”

“…In [5], the first two authors investigated existence and uniqueness of evolution systems of measures. In particular, they found that W 1 -contraction of the distance helps to prove uniqueness properties (see [5] for details). Since now the W 1 -contraction is established even in cases when the lower bound of the curvature may be negative, this allows to improve the result in [5] where a uniform lower curvature bound had been imposed for each time.…”