A partial Laplacian as an infinitesimal generator on the Wasserstein space

Abstract

In this manuscript, we consider special linear operators which we term partial Laplacians on the Wasserstein space, and which we show to be partial traces of the Wasserstein Hessian. We verify a distinctive smoothing effect of the “heat flows” they generated for a particular class of initial conditions. To this end, we will develop a theory of Fourier analysis and conic surfaces in metric spaces. We then identify a measure which allows for an integration by parts for a class of Sobolev functions. To achieve this goal, we solve a recovery problem on the set of Sobolev functions on the Wasserstein space.