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We study stochastic processes on the Wasserstein space, together with theirinfinitesimal generators. One of these processes is modeled after Brownianmotion and plays a central role in our work. Its infinitesimal generatordefines a partial Laplacian on the space of Borel probability measures, and weuse it to define heat flow on the Wasserstein space. We verify a distinctivesmoothing effect of this flow for a particular class of initial conditions. Tothis end, we will develop a theory of Fourier analysis and conic surfaces inmetric spaces. We note that the use of the infinitesimal generators has beeninstrumental in proving various theorems for Mean Field Games, and weanticipate they will play a key role in future studies of viscosity solutionsof PDEs in the Wasserstein space.

A partial Laplacian as an infinitesimal generator on the Wasserstein space