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Stochastic homogenization for Hamilton-Jacobi-Bellman equations on continuum percolation clusters

This paper establishes the almost sure stochastic homogenization of random Hamilton-Jacobi-Bellman equations on continuum percolation clusters by proving that, despite the lack of uniform ellipticity, stationarity, and finite-range dependence, the problem can be solved by leveraging the Hamiltonian's coercivity, a relative entropy structure, and the specific random geometry of the percolation cluster.

Original authors: Rodrigo Bazaes, Alexander Mielke, Chiranjib Mukherjee

Published 2026-06-11

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Rodrigo Bazaes, Alexander Mielke, Chiranjib Mukherjee

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: Navigating a Shifting Maze

Imagine you are trying to drive a car from point A to point B. In a normal city, the roads are a perfect grid: straight, predictable, and everywhere. You can easily calculate the fastest route.

Now, imagine a different world. The "roads" only exist in certain random patches. Sometimes the road is wide and smooth; other times it's a narrow, bumpy trail. Sometimes the road disappears entirely, forcing you to take a huge detour. This is a continuum percolation cluster. It's a random, messy network of connected space (like a giant, random sponge) where you can only travel through the "holes" that are connected.

The paper asks a difficult question: If you drive through this random, messy network for a very long time, does your path start to look like you are driving on a normal, smooth road?

The authors say yes. Even though the environment is chaotic, random, and full of dead ends, if you zoom out far enough, your movement averages out into a predictable, smooth pattern. They call this process homogenization.

The Characters in the Story

To understand how they proved this, let's meet the main characters:

The Driver (The Equation): The paper studies a specific math equation (Hamilton-Jacobi-Bellman or HJB) that describes how a driver makes decisions to get somewhere as fast as possible while avoiding obstacles.
The Terrain (The Percolation Cluster): This is the random environment. It's not a perfect grid. It's a "sponge" made of random points.
- The Catch: The authors are looking at a very specific, tricky version of this sponge. They are conditioning the math to say, "Assume we are already inside the giant, infinite part of the sponge."
- The Problem: Because we forced the math to start inside the giant part, the rules of the game change. The environment no longer looks the same if you shift it slightly (it's non-stationary). Also, in some spots, the "roads" are so narrow or broken that the car can't move in certain directions (it's non-elliptic or degenerate).
The Map Maker (The Effective Hamiltonian): The goal is to find a new, simple map (an "effective" equation) that describes the driver's average behavior, ignoring all the tiny, random bumps and detours.

The Challenge: Why Was This Hard?

Previous math papers solved this problem for "nice" environments where:

The roads were always smooth (uniformly elliptic).
The rules of the road were the same everywhere (stationary).

This paper tackles a much messier reality:

The "Dead End" Problem: In a random sponge, you might hit a wall where you can't move forward. The math has to handle these "dead ends" where the usual rules break down.
The "Bias" Problem: Because we forced the driver to start in the "infinite" part of the sponge, the environment looks different depending on where you are. It's like being in a forest where the trees are arranged differently if you look left vs. right, and the usual math tricks for averaging them out don't work.

The Solution: A New Way to Average

The authors used a clever method inspired by a previous technique (by Kosygina, Rezakhanlou, and Varadhan), but they had to invent new tools to handle the messiness.

1. The "Control" Analogy:
Think of the driver as having a "control" (a steering wheel). The math tries to find the best steering strategy. The authors realized that instead of looking at the driver's path directly, they could look at the "environment seen from the particle." Imagine the driver is fixed, and the world moves around them. They proved that even in this messy, shifting world, the "average" movement of the world stabilizes.

2. The "Entropy" Trick:
To handle the fact that the environment isn't uniform, they used a concept called relative entropy.

Analogy: Imagine trying to balance a stack of cards on a windy day. If the wind is random, the stack falls. But if you add a specific weight (entropy) to the bottom card, the stack becomes stable. The authors added a mathematical "weight" to their equations that forced the messy, random parts to settle down into a predictable pattern.

3. The "Corrector" (The Detour Calculator):
They introduced a concept called a "corrector."

Analogy: Imagine you are walking through a forest. You want to walk in a straight line (North). But the trees force you to zigzag. The "corrector" is a mental map that calculates exactly how much you zigzagged so you can subtract it from your total distance.
The authors proved that even in this random sponge, you can build a mental map that corrects for the detours, and as you walk further, the "zigzag" part becomes negligible compared to the straight-line distance.

The Main Result

The paper proves that:

Convergence: As the scale of the problem gets smaller (zooming in on the tiny details of the sponge) and the time gets longer, the chaotic, random solution smooths out.
The New Map: The driver's behavior can be described by a single, clean equation (the homogenized equation).
The Formula: They provided a specific formula (a variational formula) to calculate the "effective" rules of this new smooth road. This formula takes into account the specific shape of the random sponge.

Why It Matters (According to the Paper)

The paper claims this is the first time this specific type of math problem has been solved for continuum percolation clusters (random sponges) where the environment is not uniform and has dead ends.

They also show a practical application of their math: Large Deviation Principles.

Analogy: If you drop a leaf in a river with random eddies and rocks, where will it end up after a long time? Usually, it goes with the flow. But sometimes, by pure chance, it gets stuck in a whirlpool and goes the wrong way.
The authors' math allows us to calculate the exact probability of these rare, "wrong-way" events happening in a random, broken environment.

Summary in One Sentence

The authors proved that even if you are navigating a chaotic, broken, and random maze where the rules change depending on where you are, your long-term path will eventually look smooth and predictable, and they figured out exactly how to calculate that smooth path.

Technical Summary: Stochastic Homogenization of Hamilton-Jacobi-Bellman Equations on Continuum Percolation Clusters

Problem Statement
The paper addresses the stochastic homogenization of random Hamilton-Jacobi-Bellman (HJB) equations defined on the unbounded connected component (infinite cluster) of a continuum percolation model in $\mathbb{R}^d$ ( $d \geq 2$ ). Specifically, the authors study the asymptotic behavior as $\varepsilon \to 0$ of the viscosity solution $u^\varepsilon$ to the equation:
$\partial_t u^\varepsilon = \frac{\varepsilon}{2} \text{div}\left( a\left(\frac{x}{\varepsilon}, \omega\right) \nabla u^\varepsilon \right) + H\left(\frac{x}{\varepsilon}, \nabla u^\varepsilon, \omega\right),$
posed on the domain $(0, T) \times \varepsilon C_\infty(\omega)$ , where $C_\infty(\omega)$ is the infinite cluster containing the origin.

The environment presents significant challenges distinct from classical homogenization settings:

Degeneracy: The diffusion coefficient $a(x, \omega)$ is degenerate elliptic; its support is restricted to the closure of the infinite cluster, meaning the operator is not uniformly elliptic.
Non-Stationarity: The analysis is conducted under the conditional probability measure $P_0(\cdot) = P(\cdot \mid 0 \in C_\infty)$ . Unlike the unconditional measure $P$ , $P_0$ is not invariant under the translation group $\{\tau_x\}_{x \in \mathbb{R}^d}$ , breaking the standard stationarity assumptions used in prior homogenization literature.
Long-Range Correlations: The model does not assume finite-range dependence (i.i.d. structure); the percolation models may possess long-range correlations.

The Hamiltonian $H$ is assumed to be convex and coercive with respect to the semi-norm induced by the degenerate matrix $a$ , but not necessarily uniformly coercive in the Euclidean norm.

Methodology
The proof strategy is inspired by the method of Kosygina-Rezakhanlou-Varadhan [KRV06], originally developed for HJB equations in stationary, ergodic, and uniformly elliptic environments. The authors adapt this framework to the non-stationary and degenerate setting through several key technical innovations:

Optimal Control Representation: The solution $u^\varepsilon$ is represented via a variational formula involving an optimal control problem for diffusions on the percolation cluster. The authors utilize an auxiliary probability space to sample progressively measurable controls, allowing for a rigorous treatment of the "environment seen from the particle."
Ergodic Theorem for Non-Stationary Measures: A crucial step involves establishing an ergodic theorem for the environment process under the conditional measure $P_0$ . Since $P_0$ is not translation-invariant, the authors employ "induced shifts" $\sigma_e$ defined by the successive arrivals of the cluster along coordinate axes. They prove that these induced shifts preserve $P_0$ and are ergodic, enabling the application of Birkhoff's ergodic theorem to derive law of large numbers for the diffusion trajectory.
Variational Analysis and Min-Max Theorems: The authors derive a variational formula for the effective Hamiltonian. Unlike the [KRV06] approach which relies on restricting variational problems to bounded regions to ensure compactness (a technique incompatible with the non-translation-invariant $P_0$ $P_{0}$ ), this paper utilizes:
- Intrinsic Coercivity: Leveraging the coercivity of the Hamiltonian with respect to the degenerate matrix $a$ to establish weak compactness of gradients in $L^{1+\delta}(P_0)$ .
- Relative Entropy Structure: Introducing a subtractive relative entropy term in the variational analysis. This term provides the necessary coercivity to apply a second min-max theorem, facilitating the extraction of a weak limit $G$ .
Correctors and Sub-linear Growth: The existence of a "corrector" $V_G$ (a path integral of the weak limit $G$ ) is established. The authors prove that $V_G$ exhibits sub-linear growth ( $o(|x|)$ ) almost surely with respect to $P_0$ . This result relies heavily on the specific geometric properties of continuum percolation clusters (e.g., exponential decay of distances and recurrence properties) rather than combinatorial counting arguments used in discrete settings.

Key Contributions and Results

Main Theorem (Theorem 2.1): The authors prove that, $P_0$ -almost surely, the solution $u^\varepsilon$ converges uniformly on compact sets to a homogenized limit $u_{\text{hom}}$ . The limit $u_{\text{hom}}$ solves the effective equation:
$\partial_t u_{\text{hom}} = \bar{H}(\nabla u_{\text{hom}}), \quad u_{\text{hom}}(0, x) = f(x),$
where $\bar{H}$ is the effective Hamiltonian.
Variational Formula for Effective Hamiltonian: The effective Hamiltonian $\bar{H}(\theta)$ is characterized by a variational formula:
$\bar{H}(\theta) = \inf_{G \in \mathcal{G}_\delta} \left( \text{ess sup}_{P_0} \left[ \frac{1}{2} \text{div}(a(G+\theta)) + H(G+\theta) \right] \right),$
where $\mathcal{G}_\delta$ is a class of functions satisfying $L^{1+\delta}$ -boundedness, a curl-free property on the cluster, and a refined "zero induced mean" property adapted to the non-stationary measure.
Large Deviation Principle (Corollary 2.2): As a direct application, the paper establishes a quenched large deviation principle for a degenerate diffusion with random drift on a continuum percolation cluster. The rate function is given by the Legendre-Fenchel transform of the effective Hamiltonian.
Relaxation of Assumptions: The work removes the requirement for uniform ellipticity inside the cluster and the finite-range dependence (i.i.d.) assumption, allowing for models with long-range correlations.

Significance and Claims
The paper claims to be the first instance where stochastic homogenization of HJB equations has been studied on percolation clusters. The authors highlight that the inherent properties of percolation—specifically non-translation invariance and non-ellipticity—pervade the entire analysis, necessitating new mathematical tools.

The significance lies in:

Bridging Frameworks: Successfully adapting the [KRV06] method, previously limited to stationary and elliptic settings, to the non-stationary, degenerate context of continuum percolation.
Handling Non-Stationarity: Developing a rigorous framework for homogenization under the conditional measure $P_0$ , which is essential for studying the almost sure behavior of processes starting at the origin in random media.
Variational Characterization: Providing a variational representation of the effective Hamiltonian that explicitly reflects the geometric constraints of the percolation cluster (via the induced mean-zero condition and the specific function space $\mathcal{G}_\delta$ ).

The authors maintain a modest tone regarding applications, noting that while the method could potentially be extended to sub-additive approaches (as used in [So99, AT14]), their primary contribution is the establishment of the variational formula and the proof of homogenization in this specific, challenging geometric setting without assuming uniform ellipticity or i.i.d. structures.