On fully nonlinear parabolic mean field games with nonlocal and local diffusions

TL;DR

This paper studies fully nonlinear mean field games involving local and nonlocal diffusions, establishing existence and uniqueness of solutions under broad conditions, and introducing a new control interpretation related to stochastic processes.

Contribution

It introduces a new class of mean field games with a novel control interpretation and proves existence and uniqueness results for both local and nonlocal equations under general assumptions.

Findings

Existence and uniqueness of solutions for the class of mean field games.

Application to equations with fractional Laplacians and nonlocal operators.

Development of tools for analysis in the whole space without moment assumptions.

Abstract

We introduce a class of fully nonlinear mean field games posed in $[0,T]\times\mathbb{R}^d$. We justify that they are related to controlled local or nonlocal diffusions, and more generally in our setting, to a new control interpretation involving time change rates of stochastic (L\'evy) processes. The main results are existence and uniqueness of solutions under general assumptions. These results are applied to non-degenerate equations - including both local second order and nonlocal with fractional Laplacians. Uniqueness holds under monotonicity of couplings and convexity of the Hamiltonian, but neither monotonicity nor convexity need to be strict. We consider a rich class of nonlocal operators and processes and develop tools to work in the whole space without explicit moment assumptions.