Viscosity Solutions for McKean-Vlasov Control on a torus

TL;DR

This paper develops a framework for solving McKean-Vlasov control problems on a torus using viscosity solutions, introducing a new Fourier Wasserstein metric to establish uniqueness and regularity of the value function.

Contribution

It introduces a novel Fourier Wasserstein metric and applies viscosity solution theory to McKean-Vlasov control problems on a torus, establishing existence, uniqueness, and regularity results.

Findings

Value function is Lipschitz continuous under the new metric.

Comparison principle for viscosity solutions is proved.

Uniqueness of the viscosity solution is established.

Abstract

An optimal control problem in the space of probability measures, and the viscosity solutions of the corresponding dynamic programming equations defined using the intrinsic linear derivative are studied. The value function is shown to be Lipschitz continuous with respect to a novel smooth Fourier Wasserstein metric. A comparison result between the Lipschitz viscosity sub and super solutions of the dynamic programming equation is proved using this metric, characterizing the value function as the unique Lipschitz viscosity solution.