Value functions in the Wasserstein spaces: finite time horizons

摘要

We study analogs of value functions arising in classical mechanics in the space of probability measures endowed with the Wasserstein metric W-p, for 1 < p < infinity. Our main result is that each of these generalized value functions is a type of viscosity solution of an appropriate Hamilton Jacobi equation, completing a program initiated by Gangbo, Nguyen, and Tudorascu. Of particular interest is a formula we derive for a generalized value function when the associated potential energy is of the form nu(mu) = integral(Rd) V(x)d mu(x). This formula allows us to make rigorous a well known heuristic connection between Euler-Poisson equations and classical Hamilton Jacobi equations. Further results are presented which suggest there is a rich theory to be developed of deterministic control in the Wasserstein spaces. (C) 2015 Elsevier Inc. All rights reserved.