Pseudomeasure distributions for nonseparable, nonlocal mean field games

TL;DR

This paper establishes existence, uniqueness, and stability of solutions for a class of nonlocal, nonseparable mean field games with pseudomeasure initial data, expanding the mathematical understanding of such models.

Contribution

It introduces a framework for solving mean field games with nonlocal, nonseparable Hamiltonians using pseudomeasure data, including Dirac masses, and proves key properties of solutions.

Findings

Existence of solutions under small terminal data conditions.

Uniqueness and continuous dependence of solutions.

Handling of Dirac mass initial distributions in mean field games.

Abstract

For a number of important mean field games models, the Hamiltonian is non-local and not additively separable. This means that the distribution of agents appears in the Hamiltonian only in an integral over the whole spatial domain. For mean field games with a class of such Hamiltonians, we prove existence of solutions for the mean field games system of partial differential equations, allowing pseudomeasure data for the distribution of agents. Specifically, this allows the initial distribution of agents to be a sum of Dirac masses. The existence theorem requires a smallness condition on the size of the terminal data for the value function (or, alternatively, on the size of the Hamiltonian); no smallness condition on the size of the initial data or on the size of the time horizon is required. We also prove uniqueness and continuous dependence results under the same type of smallness conditions. We prove continuous dependence under two complementary hypotheses on the initial data: strong convergence of a sequence of pseudomeasures, and weak-$*$ convergence of a sequence of bounded measures.