Value functions in the Wasserstein spaces: finite time horizons

TL;DR

This paper extends the concept of value functions to Wasserstein spaces, showing they solve Hamilton-Jacobi equations and connecting them to Euler-Poisson equations, opening new avenues for deterministic control theory.

Contribution

It proves that generalized value functions in Wasserstein spaces are viscosity solutions of Hamilton-Jacobi equations and derives a formula linking these to classical mechanics.

Findings

Value functions are viscosity solutions of Hamilton-Jacobi equations in Wasserstein spaces.

Derived a formula for value functions with potential energy of a specific form.

Indications of a rich theory of deterministic control in Wasserstein spaces.

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.