Uniqueness of solutions to MFG systems with large discount

Technical Summary: Uniqueness of Solutions to MFG Systems with Large Discount

Problem Statement
The paper investigates the uniqueness of solutions to a class of Mean Field Game (MFG) systems with a large discount factor $\lambda$. The system, denoted as $(MFG_\lambda)$, consists of a coupled forward-backward system of partial differential equations (PDEs) on the space-time cylinder $\mathbb{R}^n \times (0, \infty)$:

1. A Hamilton-Jacobi-Bellman (HJB) equation for the value function $u_\lambda$:
   $$-\partial_t u_\lambda - \Delta u_\lambda + \frac{1}{\lambda}H(\lambda D u_\lambda, x) + \lambda u_\lambda = F(x, m_\lambda(t))$$
2. A Fokker-Planck (FP) equation for the population density $m_\lambda$:
   $$\partial_t m_\lambda - \Delta m_\lambda - \text{div}(m_\lambda D_p H(\lambda D u_\lambda, x)) = 0$$
   with initial condition $m_\lambda(x, 0) = m_0(x)$.

Here, $H$ is the Hamiltonian (convex in the momentum variable), $F$ is the running cost depending on the state and the population distribution, and $\lambda > 0$ serves a dual role: it is the time discount factor and is inversely proportional to the cost of controls. The authors focus on the regime where $\lambda$ is sufficiently large.

Motivation and Context
The analysis is motivated by the work of Bardi and Cardaliaguet [BC21], who demonstrated that as $\lambda \to \infty$, solutions to $(MFG_\lambda)$ converge to solutions of a nonlinear parabolic PDE $(MF_\infty)$, which is a McKean-Vlasov equation. While $(MF_\infty)$ generally admits unique solutions under standard Cauchy-Lipschitz theory, the uniqueness of solutions for finite but large $\lambda$ in the full MFG system was not established.

Existing uniqueness results in MFG theory typically rely on:

• Monotonicity structures: Specifically, Lasry-Lions monotonicity or displacement monotonicity.
• Smallness assumptions: Related to short time horizons or weak interactions.

The regime identified in this paper is distinct from these. It relies on the "large discount" parameter $\lambda$ rather than monotonicity or small time horizons. The authors note that as $\lambda$ increases, agents become less interested in future events, and the "reaction cost" becomes cheaper, effectively causing the system to degenerate toward the limit model $(MF_\infty)$.

Methodology and Preliminary Results
The authors restrict their analysis to a specific class of solutions satisfying certain growth and regularity conditions (specifically, boundedness from below and linear growth in space for $u_\lambda$, and boundedness of its derivatives).

1. A Priori Estimates (Proposition 2.1): The authors establish uniform estimates for solutions $(u_\lambda, m_\lambda)$ for large $\lambda$. Crucially, they prove that the value function $u_\lambda$ and its derivatives scale with $1/\lambda$. Notably, the constant $C$ in these estimates is independent of the diffusion parameter (viscosity), a feature highlighted as remarkable.
2. Existence (Proposition 2.2): Existence of solutions in this class is established by taking limits of solutions to the system on a finite time horizon $[0, T]$ with a zero terminal condition $u_\lambda(T) = 0$, as $T \to \infty$.
3. Stability of the Density (Proposition 2.3 & Corollary 2.4): Using a probabilistic representation involving stochastic differential equations (SDEs), the authors derive estimates for the Wasserstein distance $d_1$ between two density solutions. These estimates depend on the difference in the drift terms ($\lambda D u_\lambda$).

Key Technical Contributions
The core of the paper lies in quantifying the relationship between the gradient of the value function and the population density, and how this relationship stabilizes as $\lambda$ grows.

1. Convergence of the Drift (Proposition 3.1): The authors prove a quantitative convergence result showing that the term $\lambda D u_\lambda$ converges uniformly to $D_x F(x, m_\lambda(t))$ as $\lambda \to \infty$. Specifically:
   $$\|\lambda D u_\lambda - D_x F(\cdot, m_\lambda(\cdot))\|_\infty \lesssim \frac{1}{\lambda}$$
   This implies that for large $\lambda$, the drift driving the population evolution depends almost exclusively on the current state of the density $m_\lambda(t)$, rather than its future trajectory. This bridges the gap between the non-local-in-time nature of the MFG system and the local-in-time nature of the limit McKean-Vlasov equation.

2. Quantitative Stability of the Gradient (Proposition 3.4): A crucial step for uniqueness is establishing how the difference in gradients $\lambda D u^1_\lambda - \lambda D u^2_\lambda$ depends on the difference in densities $m^1_\lambda - m^2_\lambda$. The authors derive an estimate of the form:
   $$\|\lambda D u^1_\lambda(\cdot, t) - \lambda D u^2_\lambda(\cdot, t)\|_\infty \lesssim d_1(m^1_\lambda(t), m^2_\lambda(t)) + \int_t^\infty d_1(m^1_\lambda(r), m^2_\lambda(r)) e^{-(\lambda-\eta)(r-t)} dr$$
   This inequality indicates that for large $\lambda$, variations in the density in the "future" have exponentially decaying influence on the current gradient.

3. Uniqueness via a Non-Local Grönwall Lemma (Theorem 3.5): Combining the stability estimate for the density (Corollary 2.4) with the gradient estimate (Proposition 3.4), the authors formulate an integral inequality for the distance between two potential solutions. They employ a specialized "future-integral" Grönwall-type lemma (Lemma 2.9) to show that if $\lambda$ exceeds a certain threshold $\Lambda$, the only solution to this inequality is the zero function. Consequently, the two solutions must be identical.

Main Results
The primary result is Theorem 3.5: Under standard assumptions on the Hamiltonian $H$ and the cost function $F$ (specifically $C^{2,\alpha}$ regularity, convexity, and Lipschitz continuity), there exists a constant $\Lambda > 0$ such that for all $\lambda \geq \Lambda$, the system $(MFG_\lambda)$ admits a unique solution within the class of functions satisfying the established growth conditions.

The threshold $\Lambda$ depends on the Lipschitz constants of $F$ and $H$ but is independent of the viscosity (diffusion parameter).

Significance and Claims
The paper claims to identify a new "asymptotic uniqueness regime" that falls outside the standard monotonicity frameworks. The significance lies in:

• Borrowing Uniqueness: The proof strategy effectively "borrows" the uniqueness of the limit McKean-Vlasov equation $(MF_\infty)$ and shows it persists for finite but large $\lambda$.
• Independence from Viscosity: The result holds regardless of the diffusion strength, provided $\lambda$ is large enough.
• Structural Insight: The analysis reveals that for large $\lambda$, the non-local-in-time dependence of the MFG system becomes negligible, allowing the system to behave like a local-in-time McKean-Vlasov equation.

Limitations and Scope
The authors are modest regarding the scope of their result:

• Class of Solutions: Uniqueness is proven only for solutions satisfying specific growth conditions (Proposition 2.1). They explicitly state they cannot exclude the possibility of other solutions existing outside this class.
• Regularity of $F$: The current argument assumes $F$ is a regularizing functional of $m$ (e.g., of convolution type). The authors note that extending this to local dependence on the density $m(x,t)$ would likely require non-degenerate diffusion, which is a direction for future investigation.
• Limit Behavior: While the limit $\lambda \to \infty$ yields a unique solution, the paper does not claim that $(MFG_\lambda)$ is unique if the limit problem $(MF_\infty)$ itself has multiple solutions, though they speculate that the second-order regularization might still enforce uniqueness in such cases.

In summary, the paper provides a rigorous proof that large discount factors enforce uniqueness in MFG systems, offering a mechanism distinct from monotonicity and independent of diffusion parameters.