Abstract: We endow the space of probability measures on ${\mathbb{R}}^d$ with ${\mathrm{\Delta }}_w$, 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 ${\mathrm{\Delta }}_{w,ϵ}$, appearing in Mean Field Games theory, are considered (Joint work with Y. T. Chow).