Quantitative homogenization of convex Hamilton-Jacobi equations in the Wasserstein space

Technical Summary: Quantitative Homogenization of Convex Hamilton–Jacobi Equations in the Wasserstein Space

Problem Statement
This paper investigates the asymptotic behavior and quantitative convergence rates of first-order Hamilton–Jacobi (HJ) equations posed on the Wasserstein space of Borel probability measures with finite second moments, denoted $P_2(\mathbb{R}^d)$. The primary equation studied is:
$$\begin{cases} -\partial_t U^\varepsilon + \int_{\mathbb{R}^d} H\left(\frac{x}{\varepsilon}, x, D_m U^\varepsilon(t, m, x), m\right) m(dx) = 0, & (t, m) \in [0, T) \times P_2(\mathbb{R}^d), \\ U^\varepsilon(T, m) = G(m), & m \in P_2(\mathbb{R}^d), \end{cases}$$
where $H$ is a convex Hamiltonian exhibiting rapid spatial oscillations at scale $\varepsilon > 0$ via the fast variable $x/\varepsilon$. The solution $U^\varepsilon$ represents the value function of a mean field control problem where a large population of interacting agents minimizes a common cost functional in a heterogeneous environment.

The authors address two distinct settings based on the dependence of the Hamiltonian $H(y, x, p, m)$:

1. Non-multiscale setting (Assumption A1'): $H$ depends only on the fast variable $y = x/\varepsilon$ and the momentum $p$, independent of the macroscopic variable $x$ and the measure $m$.
2. Multiscale setting (Assumption A1): $H$ depends on the fast variable $y$, the macroscopic variable $x$, the momentum $p$, and the measure $m$.

The goal is to prove that as $\varepsilon \to 0$, $U^\varepsilon$ converges uniformly to a limit $U$ satisfying an effective HJ equation with a homogenized Hamiltonian $\bar{H}$, and to establish sharp quantitative rates for this convergence.

Methodology
The authors employ a combination of optimal control theory, viscosity solution techniques, and probabilistic methods tailored to the infinite-dimensional Wasserstein space.

1. Control Representation and Equivalence of Control Sets:
   A central technical hurdle is the lack of local compactness in $P_2(\mathbb{R}^d)$, which complicates standard doubling-of-variables arguments used in finite dimensions. To circumvent this, the authors first establish that the value function $U^\varepsilon$ remains invariant under different classes of admissible controls. They extend the standard set of progressively measurable controls $\mathcal{A}$ to larger sets $\mathcal{A}_1$ (measurable with respect to time and the filtration) and $\mathcal{A}_2$ (measurable with respect to an independent Gaussian random variable). This allows them to reformulate the stochastic control problem into a deterministic optimal transport problem involving random paths $\gamma$ connecting initial and terminal random variables $X$ and $Y$.

2. Reduction to Cost Functions:
   By exploiting the equivalence of control sets, the authors rewrite the value function $U^\varepsilon$ in terms of a running cost function $h^\varepsilon$, which represents the minimal action required to transport a particle from $x$ to $y$ over a time interval with the oscillatory Lagrangian $L$. Specifically, they show:
   $$U^\varepsilon(t, m) = \inf_{\nu \in P_2} \left( \inf_{(X,Y) \in E_{m,\nu}} \mathbb{E}[h^\varepsilon(t, T, X, Y)] + G(\nu) \right).$$
   This reduction allows the application of techniques developed for finite-dimensional homogenization (specifically those by Tran and Yu) to the infinite-dimensional setting.

3. Subadditivity and Superadditivity:
   For the non-multiscale case, the convergence rate is derived using the subadditivity and superadditivity properties of the cost function $h^\varepsilon$. These properties, inherited from the finite-dimensional results, ensure the existence of an averaged cost $h$ and provide the $O(\varepsilon)$ error bound.

4. Time-Partitioning for Multiscale Dependence:
   In the general multiscale case (where $H$ depends on $x$ and $m$), the authors cannot directly apply the $O(\varepsilon)$ argument due to the slow variables. Instead, they partition the time interval $[t, T]$ into $N$ subintervals. On each subinterval, they "freeze" the slow variables ($x$ and $m$) at their initial values, reducing the problem locally to the non-multiscale case. They then optimize the number of partitions $N$ to balance the homogenization error ($O(\varepsilon)$) and the freezing error ($O(1/N)$), resulting in an $O(\sqrt{\varepsilon})$ rate.

5. Identification of the Limit:
   Once uniform convergence is established, the authors use a perturbed test function method (doubling of variables) in the lifted Hilbert space $L^2(\Omega; \mathbb{R}^d)$ to verify that the limit $U$ is the unique $L$-viscosity solution of the effective equation. The uniqueness of viscosity solutions in the Wasserstein space (established in prior literature) ensures the identification of the limit.

Key Contributions and Results

• Optimal Rate in Non-Multiscale Setting (Theorem 1.1): Under Assumption (A1'), the authors prove that $U^\varepsilon$ converges uniformly to the solution $U$ of the effective equation with rate $O(\varepsilon)$. This matches the optimal rate known for finite-dimensional convex HJ equations depending only on the fast variable and momentum.
• Quantitative Rate in Multiscale Setting (Theorem 1.2): Under the general Assumption (A1), where the Hamiltonian depends on macroscopic variables and the measure, the authors establish a convergence rate of $O(\sqrt{\varepsilon})$. This extends the multiscale homogenization results from finite dimensions to the Wasserstein space.
• Dynamic Optimal Transport (Theorem 1.3): The results are extended to dynamic optimal transport problems where the terminal condition imposes a constraint on the final distribution (i.e., $G(m) = +\infty$ if $m \neq \nu$). The authors derive convergence rates for the transport cost $U^\varepsilon_{ot}$, showing $O(\varepsilon)$ in the simplified case and $O(\sqrt{\varepsilon})$ in the general case, with explicit dependence on the Wasserstein distance between initial and target measures.
• Corollary on Restricted Controls (Corollary 1.5): The paper demonstrates that for non-atomic initial measures, the value function remains unchanged even if the control set is restricted to controls measurable with respect to the initial state's $\sigma$-algebra. This recovers the finite-dimensional optimal rate $O(\varepsilon)$ for the classical HJ equation as a special case (when the initial measure is a Dirac mass).

Significance and Claims
The authors claim that this work provides the first quantitative convergence results for Hamilton–Jacobi equations in the Wasserstein space. While qualitative homogenization (convergence without rates) for infinite-dimensional HJ equations and dynamic optimal transport has been studied previously (e.g., by Gangbo and Tudorascu), this paper establishes sharp rates.

Specifically, the paper asserts that:

1. It extends the optimal $O(\varepsilon)$ convergence rate for convex HJ equations from finite dimensions to the infinite-dimensional Wasserstein space in the non-multiscale setting.
2. It provides the first quantitative analysis for the multiscale setting in Wasserstein space, yielding an $O(\sqrt{\varepsilon})$ rate.
3. The methodology successfully circumvents the lack of local compactness in $P_2(\mathbb{R}^d)$ by utilizing control randomization and equivalent formulations, a technique the authors note cannot be directly applied to problems involving idiosyncratic or common noise.

The paper concludes by noting that these results bridge the gap between finite-dimensional homogenization theory and mean field control/optimal transport problems in heterogeneous environments, providing a rigorous foundation for the effective dynamics of large interacting systems.