Regularity for second order stationary mean-field games

Abstract

In this paper, we prove the existence of classical solutions for second order stationary mean-field game systems. These arise in ergodic (mean-field) optimal control, convex degenerate problems in calculus of variations, and in the study of long-time behavior of time-dependent mean-field games. Our argument is based on the interplay between the regularity of solutions of the Hamilton-Jacobi equation in terms of the solutions of the Fokker-Planck equation and vice-versa. Because we consider different classes of couplings, distinct techniques are used to obtain a priori estimates for the density. In the case of polynomial couplings, we recur to an iterative method. An integral method builds upon the properties of the logarithmic function in the setting of logarithmic nonlinearities. This work extends substantially previous results by allowing for more general classes of Hamiltonians and mean-field assumptions.

Stationary focusing mean-field games

“…where T N k = {x ∈ R N : x k + a k x ∈ T N } and H k (p), Λ k , W k (x), F k (µ) are as in (33). Local a-priori estimates on ∇v k and µ k provide Br µ α+1 k dx ≥ c > 0 (see (34)), which contradicts…”

“…A serious technical difficulty here comes from the fact that the couplings −f we consider are not bounded from below. The particular coupling −f (m) = log m, having the opposite monotonicity but lacking of boundedness from below, has been treated for example in [34,16] (see also references therein). Still, we consider this framework defocusing, as log(·) is increasing.…”

Proximal Methods for Stationary Mean Field Games with Local Couplings

“…In order to introduce the system we study, let T n be the n-dimensional torus and f : T n × [0, +∞[→ R be a continuous function. Given ν ≥ 0 and a function H : T n × R n → R, such that for all x ∈ T n the function H(x, ·) : p → H(x, p) is convex and differentiable, we consider the following stationary MFG problem: find two functions u, m and λ ∈ R such that When ν > 0, well-posedness of system (1.1) has been studied in several articles, starting with the works by J.-M. Lasry and P.-L. Lions [51,53], followed by [45,46,33,43,7,61] in the case of smooth solutions and [33,11,39,34] in the case of weak solutions. Let us point out that in terms of the underlying game, system (1.1) involves local couplings because the right hand side of (1.1) depends on the distribution m through its pointwise value (see [53]).…”

Explicit solutions of one-dimensional, first-order, stationary mean-field games with congestion

“…For second-order problems, existence of stationary solutions was proven in [6]. In the stationary setting, existence and regularity of MFGs without congestion were considered in [10], [12], [14], [22], [23] (strong solutions) and [5], [19] (weak solutions). Many other stationary problems are examined in the literature, including obstacle problems [9], weaklycoupled systems [8], multi-populations [4], and logistic problems [13].…”