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Math Colloquium
Yat Tin Chow Univ. of California, Los Angeles Title: A partial Laplacian as an infnitesimal generator on Wasserstein space and a numerical method towards Hamilton-Jacobi equations on Wasserstein space Abstract: In this talk we discuss two topics.In the first part of the 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. 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 part formula, Ito formula and a 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 spaceIn the second part of the talk, we explore possibility of computing solutions of a certain type of infinitely dimensional Hamilton-Jacobi equations in probability space that arises in optimal control on Wasserstein space and the theory of mean field games. We propose to utilize a Hopf-type formula coming from an optimal control approach. The resulting formula is an optimization problem involving a Cauchy problem of d dimensional HJ-PDE constraint (the mean field equations) which can be computed using a standard finite difference scheme. They may be of importance in computing optimization problems involving Wasserstein metrics. Our techniques may have applications in optimal transport, mean field games and optimal control in the space of probability densities. Date: Monday, December 4, 2017Time: 4:30 pmLocation: Wean Hall 7218Submitted by: BohmanNote: Refreshments at 4:00 pm, Wean Hall 6220. |