Value functions in the Wasserstein spaces: finite time horizons

Ryan Hynd; Hwa Kil Kim

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arXiv · 1307.4667

Value functions in the Wasserstein spaces: finite time horizons

Ryan Hynd·Hwa Kil Kim

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Abstract

We study analogs of value functions arising in classical mechanics in the space of probability measures endowed with the Wasserstein metric $W_p$, for $1<p<\infty$. 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, Tudorascu, and Nguyen. Of particular interest is a formula we derive for a generalized value function when the associated potential energy is of the form ${\cal V}(\mu)=\int_{\mathbb{R}^d}V(x)d\mu(x)$. 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.

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Ryan Hynd, Hwa Kil Kim. 2013-07-17. Value functions in the Wasserstein spaces: finite time horizons. https://arxiv.org/abs/1307.4667

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