Two Remarks on the Wasserstein Dirichlet Form

Abstract

The Wasserstein diffusion is an Ornstein–Uhlenbeck type process on the set of all probability measures with the Wasserstein metric as intrinsic metric. Sturm and von Renesse constructed in [6] this process in the case of probability measures over the unit interval using Dirichlet form theory. An essential step in this construction is the closability of a certain gradient form, defined for smooth cylindrical test functions, in the space L ² w.r.t. the entropic measure ℚ_β. In this paper we will first give an alternative proof for this closability, avoiding the striking, but elaborate integration by parts formula for ℚ_β used in [6]. Second, we give explicit conditions under which certain finite-dimensional particle approximations introduced in the paper [1] by Andres and von Renesse do converge in the resolvent sense to the Wasserstein diffusion, a question that was left open in the above cited paper.

Mathematics Subject Classification (2010). Primary: 58J65, Secondary: 47D07, 60J35, 60K35.