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Degenerate Diffusions on Continuum Percolation and Hamilton-Jacobi-Bellman Equations

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Pith's one-line read Under suitable assumptions, controlled diffusions on continuum percolation clusters never reach the random boundary and admit unique global strong solutions.

desk verdict The paper shows that controlled degenerate diffusions on continuum percolation clusters stay away from the boundary under explicit assumptions, yielding a boundary-free representation for HJB equations on the random geometry. read the letter →

arxiv 2606.09691 v1 pith:FGFP7SWG submitted 2026-06-08 math.PR

Rodrigo Bazaes , Alexander Mielke , Chiranjib Mukherjee This is my paper · ORCID
classification math.PR
keywords degeneratediffusionscontinuumpercolationHamilton-Jacobi-BellmanequationsviscositysolutionsstochasticcontrolBooleanmodelquenchedframeworkrandomclusters
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper shows that degenerate controlled diffusions constrained to infinite continuum percolation clusters, with degeneracy tied to distance from the irregular boundary, stay inside the cluster under regularity conditions on the diffusion matrix. This yields unique global strong solutions to the associated stochastic differential equations. The same property produces a stochastic control representation for viscosity solutions of the corresponding degenerate Hamilton-Jacobi-Bellman equations on the random cluster, without any boundary conditions. The required integrability and degeneracy assumptions are verified explicitly for the Boolean model, establishing a quenched setting that connects the diffusion's analytic structure directly to the cluster geometry.

What carries the argument

Controlled diffusion whose degeneracy is governed by the distance to the random boundary of the infinite percolation cluster.

What would settle it

A simulation of the controlled diffusion on a realized Boolean model cluster in which paths reach the boundary under the paper's stated degeneracy conditions would falsify the global existence claim.

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Extended reading notes

Core claim

Under suitable regularity and degeneracy assumptions on the diffusion matrix, the corresponding controlled diffusion never reaches the boundary and therefore admits a unique global strong solution; this yields a stochastic control representation formula for viscosity solutions of degenerate HJB equations posed on the random cluster without imposing boundary conditions. The structural assumptions, including quantitative integrability properties of the distance-to-the-boundary function and the admissible degeneracy regime, hold for concrete models such as the Boolean model.

Load-bearing premise

The quantitative integrability properties of the distance-to-the-boundary function and the admissible degeneracy regime must hold for the specific percolation model.

Editorial extensions

If this is right

  • The stochastic differential equation has a unique global strong solution that remains inside the cluster for all time.
  • Viscosity solutions of the degenerate HJB equation on the random cluster admit an explicit stochastic control representation without boundary conditions.
  • The framework applies directly to quenched stochastic control problems on genuine continuum percolation geometries.
  • The boundary geometry enters the theory only through the admissible degeneracy regime rather than through explicit boundary conditions.
  • The results supply the analytic foundation for a homogenization theory on the same random clusters.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same distance-based degeneracy construction could be tested on other continuum percolation models with comparable integrability, such as random geometric graphs.
  • The absence of boundary conditions suggests that numerical schemes for the HJB equations can be implemented on truncated but interior-only domains.
  • The link between diffusion degeneracy and cluster geometry may extend to controlled processes on discrete percolation graphs with analogous distance functions.
  • Numerical verification of path non-exit on large finite clusters would provide direct evidence for the global solution property.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

0 major / 3 minor

Summary. The manuscript develops a theory of degenerate controlled diffusions constrained to infinite clusters arising in continuum percolation. Under regularity and degeneracy assumptions on the diffusion matrix (degenerating with distance to the random boundary), the controlled process is shown never to attain the boundary and therefore to admit a unique global strong solution. This non-attainability is used to derive a stochastic control representation formula for viscosity solutions of the associated degenerate HJB equations posed on the cluster, without any boundary conditions. The structural assumptions, including quantitative integrability properties of the distance-to-boundary function, are verified explicitly for the Boolean model. The results supply a quenched framework linking the analytic structure of the diffusion to the stochastic geometry of the cluster and serve as the foundation for homogenization theory in the companion paper [BMM26].

Significance. If the results hold, the work is significant because it supplies the first rigorous quenched stochastic-control representation for degenerate HJB equations on genuine random percolation geometries, together with an explicit verification of the required integrability and degeneracy conditions for a standard model (Boolean). The avoidance of boundary conditions and the direct use of the percolation geometry as the source of degeneracy are technically noteworthy and directly enable the homogenization program announced in the companion article.

minor comments (3)
  1. In the introduction and the statement of the main representation theorem, make explicit the precise functional form of the admissible degeneracy regime (i.e., the precise dependence of the diffusion coefficients on dist(·,∂C)) so that the reader can immediately check compatibility with the integrability assumption.
  2. The notation for the random cluster, its boundary, and the distance function should be introduced once in a dedicated preliminary subsection and then used consistently; occasional re-definition of symbols interrupts readability.
  3. Add a short remark after the verification for the Boolean model clarifying whether the same constants work uniformly for all realizations or only almost surely; this affects the quenched nature of the subsequent homogenization results.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript, the accurate summary of its contributions, and the recommendation of minor revision. The report correctly identifies the core results: unique global strong solutions for controlled degenerate diffusions on infinite percolation clusters under quantitative degeneracy assumptions, the resulting boundary-free stochastic control representation for the associated HJB equations, and the explicit verification of the structural assumptions for the Boolean model. These elements establish the quenched framework needed for the homogenization theory in the companion paper.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity detected

full rationale

The paper's central results establish global strong solutions for controlled degenerate diffusions on percolation clusters and a stochastic control representation for viscosity solutions of the associated HJB equations, both derived directly from structural assumptions on the diffusion matrix degeneracy and quantitative integrability of the distance-to-boundary function. These assumptions are introduced as hypotheses and then verified explicitly for the Boolean model; the representation formula follows from the existence theorem without any reduction to fitted parameters, self-definitional loops, or load-bearing self-citations. The companion article [BMM26] is referenced solely as a downstream application for homogenization and does not supply any premise used in the present derivations. The logical chain is therefore self-contained against external benchmarks.

Assumptions & free parameters 0 free parameters · 2 assumptions · 0 invented entities

The central claims rest on unverified quantitative integrability of the distance-to-boundary function and an admissible degeneracy regime whose precise form is not given in the abstract; these function as domain assumptions rather than derived properties.

assumptions (2)
  • domain assumption Quantitative integrability properties of the distance-to-the-boundary function hold for the percolation cluster
    Invoked to guarantee the diffusion never reaches the boundary; stated as a structural assumption verified for the Boolean model.
  • domain assumption The diffusion matrix satisfies suitable regularity and degeneracy assumptions compatible with the distance function
    Required for global existence of the strong solution and for the stochastic control representation.

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Cite this review

Pith. "Pith review of Degenerate Diffusions on Continuum Percolation and Hamilton-Jacobi-Bellman Equations." pith.science (2026). https://pith.science/paper/FGFP7SWG

@misc{pith2026260609691,
  author       = {Pith},
  title        = {Pith review of: Degenerate Diffusions on Continuum Percolation and Hamilton-Jacobi-Bellman Equations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FGFP7SWG}},
  note         = {Machine review of arXiv:2606.09691}
}
read the original abstract

We study degenerate controlled diffusions and Hamilton--Jacobi--Bellman equations posed on genuine continuum percolation clusters. The diffusion is constrained to evolve inside the infinite cluster and degenerates according to the distance to the irregular random boundary. Under suitable regularity and degeneracy assumptions on the diffusion matrix, we prove that the corresponding controlled diffusion never reaches the boundary and therefore admits a unique global strong solution. Using this result, we establish a stochastic control representation formula for viscosity solutions of degenerate Hamilton--Jacobi--Bellman equations posed on the random cluster, without imposing boundary conditions. We further verify that the structural assumptions introduced in this work, including quantitative integrability properties of the distance-to-the-boundary function and the associated degeneracy, hold for concrete continuum percolation models such as the Boolean model. In particular, although the diffusion never reaches the boundary, the boundary geometry remains a fundamental ingredient of the theory through the admissible degeneracy regime. The results establish a quenched framework for stochastic control and degenerate partial differential equations on genuine continuum percolation geometries, and identify a link between the analytic structure of the diffusion and the stochastic geometry of the underlying cluster. They also provide the foundational framework for the homogenization theory developed in the companion article [BMM26].

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