  We study stochastic processes on the Wasserstein space, together with their
infinitesimal generators. One of these processes is modeled after Brownian
motion and plays a central role in our work. Its infinitesimal generator
defines a partial Laplacian on the space of Borel probability measures, and we
use it to define heat flow on the Wasserstein space. We verify a distinctive
smoothing effect of this flow 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 note that the use of the infinitesimal generators has been
instrumental in proving various theorems for Mean Field Games, and we
anticipate they will play a key role in future studies of viscosity solutions
of PDEs in the Wasserstein space.
