Georgia Institute of Technology College of Sciences

In this talk, we will introduce a family of 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, taken as a partial trace of a Hessian. We study the eigenfunction of this partial Laplacian and develop a theory of Fourier analysis. We also consider the heat flow generated by this partial Laplacian on the Wasserstein space, and discuss smoothing effect of this flow for a particular class of initial conditions. Integration by parts formula, Ito formula and an analogous Feynman-Kac formula will be discussed. We note the use of the infinitesimal generators in the theory of Mean Field Games, and we expect they will play an important role in future studies of viscosity solutions of PDEs in the Wasserstein space.