Mild and weak solutions of Mean Field Games problem for linear control systems

2019/07/04 by Piermarco Cannarsa, Cannarsa, Piermarco, Cristian Mendico +1 · 3 citations
Computer Science · Earth and Planetary Sciences · Economics, Econometrics and Finance · Engineering · Mathematics · #Aquatic and Environmental Studies #FOS: Mathematics #Guidance and Control Systems #Nonlinear Differential Equations Analysis #Optimization and Control (math.OC) #Optimization and Variational Analysis #Stochastic processes and financial applications

paper · pdf · doi:10.48550/arxiv.1907.02654

openalex publication_date 2019/07/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

The aim of this paper is to study first order Mean field games subject to a linear controlled dynamics on \mathbb R^d. For this kind of problems, we define Nash equilibria (called Mean Field Games equilibria), as Borel probability measures on the space of admissible trajectories, and mild solutions as solutions associated with such equilibria. Moreover, we prove the existence and uniqueness of mild solutions and we study their regularity: we prove Hölder regularity of Mean Field Games equilibria and fractional semiconcavity for the value function of the underlying optimal control problem. Finally, we address the PDEs system associated with the Mean Field Games problem and we prove that the class of mild solutions coincides with a suitable class of weak solutions.

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