[Paper Review] Local regularity for mean-field games in the whole space

Diogo A. Gomes, Edgard A. Pimentel|arXiv (Cornell University)|Jul 3, 2014

Stochastic processes and financial applications6 references20 citations

TL;DR

This paper establishes local Sobolev regularity for solutions to time-dependent mean-field games in the whole space ℝᵈ by introducing a novel entropy dissipation estimate for the adjoint variable, which enables uniform bounds in W¹ᵖ_loc(ℝᵈ) for the Hamilton-Jacobi equation via the non-linear adjoint method. The key contribution is a local L∞ estimate for the spatial gradient Du in terms of the Lᶜ(0,T;Lᵃ(ℝᵈ)) norm of the non-linearity g, under suitable assumptions on the Hamiltonian and non-linearity.

ABSTRACT

In this paper, we investigate the Sobolev regularity for mean-field games in the whole space $\Rr^d$. This is achieved by combining integrability for the solutions of the Fokker-Planck equation with estimates for the Hamilton-Jacobi equation in Sobolev spaces. To avoid the mathematical challenges posed due to the lack of compactness, we prove an entropy dissipation estimate for the adjoint variable. This, together with the non-linear adjoint method, yields uniform estimates for solutions of the Hamilton-Jacobi equation in $W^{1,p}_{loc}(\Rr^d)$.

Motivation & Objective

• To establish local Sobolev regularity for solutions to time-dependent mean-field games in the whole space ℝᵈ.
• To overcome the lack of compactness in ℝᵈ, which invalidates standard regularity estimates and prevents direct application of the adjoint method.
• To develop a new entropy dissipation estimate for the adjoint variable to control the growth of solutions in Sobolev spaces.
• To derive uniform local estimates for the gradient Du in W¹ᵖ_loc(ℝᵈ) in terms of the Lᶜ(0,T;Lᵃ(ℝᵈ)) norm of the non-linearity g.
• To extend the applicability of the adjoint method beyond bounded domains by incorporating integrability and interpolation techniques in unbounded settings.

Proposed method

• Introduce an entropy dissipation estimate for the adjoint variable to control the growth of solutions in unbounded domains.
• Apply the non-linear adjoint method to the Hamilton-Jacobi equation, using the adjoint variable to derive a priori estimates.
• Use Hölder and Gagliardo-Nirenberg inequalities to relate norms of the adjoint variable to those of the non-linearity g and its derivatives.
• Employ interpolation techniques and cut-off functions to localize estimates in space and time, focusing on balls 𝒟_R.
• Derive a system of inequalities involving exponents P, Q, a′, c′, β, κ to satisfy the required embedding and integrability conditions.
• Apply Young’s inequality with ε-weighting to bound the Lᶜ′(0,T;Lᵃ′(ℝᵈ)) norm of the adjoint variable in terms of the Lᶜ(0,T;Lᵃ(ℝᵈ)) norm of g.

Experimental results

Research questions

• RQ1 How can local Sobolev regularity be established for mean-field game systems in the whole space ℝᵈ, where standard compactness arguments fail?
• RQ2 What role does entropy dissipation play in controlling the growth of solutions to the adjoint equation in unbounded domains?
• RQ3 Can the non-linear adjoint method be adapted to yield local W¹ᵖ_loc regularity estimates when the Hamiltonian is not integrable?
• RQ4 What conditions on the non-linearity g and the Hamiltonian H ensure that the gradient Du remains bounded in L∞(0,T;Lᵖ(ℬ_R)) for any R > 0?
• RQ5 How can the interplay between integrability of m, g, and the adjoint variable be exploited to derive uniform local estimates in Sobolev spaces?

Key findings

• The paper establishes a local L∞ estimate for the spatial gradient Du in terms of the Lᶜ(0,T;Lᵃ(ℝᵈ)) norm of the non-linearity g, with the bound depending only on R and the data.
• An entropy dissipation estimate for the adjoint variable is derived, which is essential for overcoming the lack of compactness in ℝᵈ.
• The non-linear adjoint method is successfully extended to unbounded domains by combining it with entropy estimates and interpolation techniques.
• The existence of exponents P, Q, a′, c′, β, κ satisfying (24)–(27) is proven, enabling the application of Hölder and Gagliardo-Nirenberg inequalities.
• The main result, Theorem 1.1, shows that for every R > 0, there exists a constant C_R such that ‖Du‖_L∞(0,T;Lᵖ(ℬ_R)) ≤ C_R, under Assumptions A1–A9.
• The proof relies on a priori estimates that allow for subsequent higher-order regularity via standard methods, as noted in the paper’s concluding remarks.