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A partial Laplacian as an infinitesimal generator on the Wasserstein space

Banff International Research Station for Mathematical Innovation and Discovery

We study stochastic processes on the Wasserstein space, together with their infinitesimal generators. One of these processes plays a central role in our work. Its infinitesimal generator defines a partial Laplacian on the space of Borel probability measures, and we use it to define heat flow on the Wasserstein space. We verify a distinctive smoothing effect of this flow for a particular class of initial conditions. To this end, we will develop a theory of Fourier analysis and conic surfaces in metric spaces. We note that the use of the infinitesimal generators has been instrumental in proving various theorems for Mean Field Games, and we anticipate they will play a key role in future studies of viscosity solutions of PDEs in the Wasserstein space (Joint work with Y. T. Chow).

Mathematics; Partial differential equations; Statistical mechanics, structure of matter

video/mp4

Author affiliation: UCLA

2019-03-16

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/68807

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/