Regularity for Second-Order Stationary Mean-Field Games

Author(s)

Pimentel, Edgard A. ^[1] ; Voskanyan, Vardan ^{[2, 3]}

Affiliation

^[1] Pontificia Univ Catolica Rio de Janerio PUC Rio, Dept Math, BR-22451900 Gavea, RJ - Brazil

^[2] KAUST, CEMSE Div, Thuwal 239556900 - Saudi Arabia

^[3] Kaust Sri Ctr Uncertainty Quantificat Computat Sc, Thuwal - Saudi Arabia

• Document type: Journal article
• Source: Indiana University Mathematics Journal; v. 66, n. 1, p. 1-22, 2017.
• Web of Science Citations: 9

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 resort 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. (AU)

• FAPESP's process: 15/13011-6 - Nonlinear Partial Differential Equations: Well-Posedness and Regularity Theory
• Grantee: Edgard Almeida Pimentel
• Support Opportunities: Regular Research Grants