[Paper Review] Existence of weak solutions to time-dependent mean-field games

Rita Ferreira, Diogo A. Gomes+1 more|arXiv (Cornell University)|Jan 12, 2020

Stochastic processes and financial applications26 references4 citations

TL;DR

This paper establishes the existence of weak solutions to a broad class of time-dependent mean-field games (MFGs) with nonlocal coupling and degenerate parabolic dynamics. Using a high-order space-time elliptic regularization and Schaefer’s fixed-point theorem, the authors prove existence and uniqueness for the regularized problem, then apply Minty’s method to pass to the limit and obtain a weak solution to the original MFG system under general monotonicity and regularity conditions on the coupling and Hamiltonian terms.

ABSTRACT

Here, we establish the existence of weak solutions to a wide class of time-dependent monotone mean-field games (MFGs). These MFGs are given as a system of degenerate parabolic equations with initial and terminal conditions. To construct these solutions, we consider a high-order elliptic regularization in space-time. Then, using Schaefer's fixed-point theorem, we obtain the existence and uniqueness for this regularized problem. Using Minty's method, we prove the existence of a weak solution to the original MFG. Finally, the paper ends with a discussion on congestion problems and density constrained MFGs.

Motivation & Objective

• To establish the existence of weak solutions for a wide class of time-dependent mean-field games with nonlocal coupling and degenerate parabolic dynamics.
• To address the challenge of solving MFG systems with general monotone coupling terms and non-smooth data by introducing a high-order space-time regularization.
• To extend the theory of MFGs to cases with initial and terminal conditions, nonlocal interactions, and potential congestion effects.
• To analyze density-constrained MFGs where the agent density is bounded above, modeling physical or social constraints on population distribution.

Proposed method

• Introduce a high-order space-time elliptic regularization to the original degenerate parabolic MFG system, transforming it into a well-posed, uniformly parabolic problem.
• Apply Schaefer’s fixed-point theorem to prove existence and uniqueness of solutions to the regularized problem in Sobolev-type function spaces.
• Use Minty’s method to pass to the limit as the regularization parameter tends to zero, establishing convergence of the regularized solutions to a weak solution of the original MFG system.
• Employ variational formulations and weak formulations of the Hamilton–Jacobi and Fokker–Planck equations to handle the nonlinearity and degeneracy.
• Define appropriate function spaces with constraints (e.g., $L^1$ for density, $L^ ho$ for value function) and use compactness and weak convergence arguments.
• Handle density constraints by introducing a bounded set $\widehat{\mathcal{A}}_1$ of admissible densities and adapting the fixed-point argument to this constrained setting.

Experimental results

Research questions

• RQ1 Under what conditions does a weak solution exist for time-dependent mean-field games with nonlocal coupling and degenerate diffusion?
• RQ2 Can the existence of solutions be established for MFGs with general monotone coupling terms and non-smooth data?
• RQ3 How can the solution theory be extended to include density constraints, such as $0 \leq m \leq M$?
• RQ4 What role does high-order space-time regularization play in proving existence for degenerate parabolic MFG systems?
• RQ5 Can Minty’s method be effectively applied to time-dependent MFGs with terminal conditions and nonlocal interactions?

Key findings

• The authors prove the existence of a weak solution $(m, \tilde{u}) \in L^1(\Omega_T) \times L^\gamma((0,T); W^{1,\gamma}(\mathbb{T}^d))$ to the original time-dependent MFG system under general monotonicity and regularity assumptions.
• The solution satisfies $m \geq 0$ and $m \leq M$ almost everywhere in $\Omega_T$ when density constraints are imposed.
• The regularized problem admits a unique solution for each $\epsilon > 0$, and this solution converges weakly to a weak solution of the original system as $\epsilon \to 0$.
• The convergence is established using compactness arguments based on Morrey’s theorem and the Rellich–Kondrachov embedding, ensuring strong convergence in $C^{2,l}(\overline{\Omega}_T)$ for some $l \in (0,1)$.
• The method applies to MFGs with nonlocal coupling $g(m, h(\boldsymbol{m}))$, where $h$ is a nonlinear operator on the measure-valued density $\boldsymbol{m}$, extending the scope beyond local couplings.
• The existence result holds even when the Hamiltonian $H$ is not strictly convex or coercive, provided the growth conditions on $g$ and $H$ are satisfied.