In this talk, we will consider special linear operators which we term partial Laplacians on the Wasserstein space, and which we show to be partial traces of the Wasserstein Hessian. We verify a smoothing effect of the "heat flows" they generated for a particular class of initial conditions. To this end, we will develop a theory of Fourier analysis and conic surfaces in metric spaces and discuss a class of Sobolev functions. To achieve this goal, we solve a recovery problem on the set of Sobolev functions on the Wasserstein space. We remark that this infinitesimal generators is closely related to the common noise in mean field games, and we anticipate they will play a role in future studies of viscosity solutions of PDEs in the Wasserstein space.
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