Singular Perturbations of Hamilton-Jacobi Equations in the Wasserstein Space

TL;DR

This paper investigates the limiting behavior of second-order Hamilton-Jacobi equations in the Wasserstein space as a small perturbation parameter approaches zero, using viscosity solutions and connecting to stochastic differential equations.

Contribution

It introduces a novel analysis of singular perturbations for Hamilton-Jacobi equations in the Wasserstein space, extending viscosity solution methods to this setting.

Findings

Characterizes solution behavior as perturbation parameter tends to zero

Adapts the perturbed test function method to Wasserstein space

Establishes a connection with homogenization of McKean-Vlasov equations

Abstract

We study a singular perturbation problem for second-order Hamilton-Jacobi equations in the Wasserstein space. Specifically, we characterize the behavior of the solutions as the perturbation parameter $\varepsilon$ tends to zero. The notion of solution we adopt is that of viscosity solutions in the sense of test functions on the Wasserstein space. Our proof utilizes the perturbed test function method, appropriately adapted to this setting. Finally, we highlight a connection with the homogenization of conditional slow-fast McKean-Vlasov stochastic differential equations.