Viscosity Solutions of the Eikonal Equation on the Wasserstein Space

TL;DR

This paper extends viscosity solution theory to Eikonal equations on the Wasserstein space, using Fourier Sobolev norms and comparison principles to analyze mean field control problems.

Contribution

It introduces a novel approach combining Fourier representation and classical techniques to establish comparison results for viscosity solutions on the Wasserstein space.

Findings

Comparison result for sub and super solutions

Lipschitz continuity of the value function verified

Point-wise upper bound for derivatives established

Abstract

Dynamic programming equations for mean field control problems with a separable structure are Eikonal equations on the Wasserstein space. Standard differentiation using linear derivatives yield a direct extension of the classical viscosity theory. We use Fourier representation of the Sobolev norms on the space of measures, together with the standard techniques from the finite dimensional theory to obtain a comparison result among sub and super solutions that are Lipschitz continuous in the Wasserstein-one norm which is directly verified for the value function. A key technical result provides a point-wise upper bound for the derivative of the linear derivative of a function in terms of its Lipschitz norm.