409 research outputs found
Stability of the global attractor under Markov-Wasserstein noise
We develop a "weak Wa\.zewski principle" for discrete and continuous time
dynamical systems on metric spaces having a weaker topology to show that
attractors can be continued in a weak sense. After showing that the Wasserstein
space of a proper metric space is weakly proper we give a sufficient and
necessary condition such that a continuous map (or semiflow) induces a
continuous map (or semiflow) on the Wasserstein space. In particular, if these
conditions hold then the global attractor, viewed as invariant measures, can be
continued under Markov-type random perturbations which are sufficiently small
w.r.t. the Wasserstein distance, e.g. any small bounded Markov-type noise and
Gaussian noise with small variance will satisfy the assumption.Comment: 19 page
Uniformly convex metric spaces
In this paper the theory of uniformly convex metric spaces is developed.
These spaces exhibit a generalized convexity of the metric from a fixed point.
Using a (nearly) uniform convexity property a simple proof of reflexivity is
presented and a weak topology of such spaces is analyzed. This topology called
co-convex topology agrees with the usualy weak topology in Banach spaces. An
example of a -spaces with weak topology which is not Hausdorff is
given. This answers questions raised by Monod 2006, Kirk and Panyanak 2008 and
Esp\'inola and Fern\'andez-Le\'on 2009.
In the end existence and uniqueness of generalized barycenters is shown and a
Banach-Saks property is proved.Comment: 23 page
On Interpolation and Curvature via Wasserstein Geodesics
In this article, a proof of the interpolation inequality along geodesics in
-Wasserstein spaces is given. This interpolation inequality was the main
ingredient to prove the Borel-Brascamp-Lieb inequality for general Riemannian
and Finsler manifolds and led Lott-Villani and Sturm to define an abstract
Ricci curvature condition. Following their ideas, a similar condition can be
defined and for positively curved spaces one can prove a Poincar\'e inequality.
Using Gigli's recently developed calculus on metric measure spaces, even a
-Laplacian comparison theorem holds on -infinitesimal convex spaces. In
the appendix, the theory of Orlicz-Wasserstein spaces is developed and
necessary adjustments to prove the interpolation inequality along geodesics in
those spaces are given.Comment: 35+14 pages, comments are welcome, added remark on relation between
weak and strong curvature conditio
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