A priori estimates and large population limits for some nonsymmetric Nash systems with semimonotonicity

TL;DR

This paper establishes uniform derivative estimates for solutions to semilinear parabolic systems modeling N-player differential games, enabling analysis of large population limits in heterogeneous Mean Field Games without symmetry assumptions.

Contribution

It provides novel uniform estimates on derivatives of solutions to nonsymmetric Nash systems, facilitating the study of large population limits in heterogeneous Mean Field Games.

Findings

Derivatives of solutions are uniformly bounded in N.

Results apply to systems with non-symmetric data.

Addresses convergence as N approaches infinity.

Abstract

We address the problem of regularity of solutions $u^i(t, x^1, \dots, x^N)$ to a family of semilinear parabolic systems of $N$ equations, which describe closed-loop equilibria of some $N$-player differential games with Lagrangian having quadratic behaviour in the velocity variable, running costs $f^i(x)$ and final costs $g^i(x)$. By global (semi)monotonicity assumptions on the data $f=(f^i)_{1 \leq i \leq N}$ and $g=(g^i)_{1 \leq i \leq N}$, and assuming that derivatives of $f^i, g^i$ in directions $x^j$ are of order $1/N$ for $j \neq i$, we prove that derivatives of $u^i$ enjoy the same property. The estimates are uniform in the number of players $N$. Such a behaviour of the derivatives of $f^i, g^i$ arise in the theory of Mean Field Games, though here we do not make any symmetry assumption on the data. Then, by the estimates obtained we address the convergence problem $N \to \infty$ in a "heterogeneous'' Mean Field framework, where players all observe the empirical measure of the whole population, but may react differently from one another. We also discuss some results on the joint $N \to \infty$ and vanishing viscosity limit.