Laplacian operator and Brownian motions on the Wasserstein space

We endow the space of probability measures on $\mathbb{R}^d$ with $\Delta_w$, a Laplacian operator. A Brownian motion is shown to be consistent with the Laplacian operator. The smoothing effect of the heat equation is established for a class of functions. Special perturbations of the Laplacian operator, denoted $\Delta_{w,\epsilon}$, appearing in Mean Field Games theory, are considered (Joint work with Y. T. Chow).