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

We address the problem of regularity of solutions uⁱ(t, x¹, …, xN) to a family of semilinear parabolic systems of N equations, which describe closed-loop equilibria of some N-player differential games with quadratic Lagrangian in the velocity, running costs fⁱ(x) and final costs gⁱ(x). By global (semi)monotonicity assumptions on the data $f,g$, and assuming that derivatives of fⁱ, gⁱ in directions xⱼ are of order $1/N$ for j ≠ i, we prove that derivatives of uⁱ enjoy the same property. The estimates are uniform in the number of players N. Such behaviour of the derivatives of fⁱ, gⁱ 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 → ∞ in a ``heterogeneous'' Mean Field framework, where players all observe the empirical measure of the whole population, but may react differently from one another.