arxiv: 2602.16126 · v4 · submitted 2026-02-18 · 🧮 math.PR · math.DS

Martin Boundary and the Nonlinear Multiplicative Stochastic Heat Equation in Weak Disorder

Hongyi Chen This is my paper

Pith reviewed 2026-05-15 21:47 UTC · model grok-4.3

classification 🧮 math.PR math.DS

keywords invariant random fieldsstochastic heat equationweak disorderMartin boundaryharmonic functionsmultiplicative noisepositive invariants

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The pith

Positive invariant fields of the nonlinear multiplicative stochastic heat equation correspond one-to-one with bounded positive harmonic functions on the space.

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

The paper shows that under a second-moment condition, the positive invariant random fields for these equations stand in bijection with the bounded positive harmonic functions of the underlying space. This bijection transfers the Martin boundary structure directly to the space of invariant fields, classifying them via the geometry of the domain. It further proves that stochastic convergence to an invariant field follows whenever the deterministic heat flow converges to the matching harmonic function. The correspondence holds in settings with rich Martin boundaries, such as negatively curved manifolds and trees. This links long-time stochastic behavior in weak disorder to classical potential theory without introducing new random structures.

Core claim

Positive invariant fields are in one-to-one correspondence with bounded positive harmonic functions of the underlying space. This implies that the space of invariant fields inherits the structure of the Martin boundary. Whenever the deterministic heat flow converges to a bounded harmonic function, the stochastic evolution converges to the corresponding invariant field. The results apply to many settings with nontrivial Martin boundary, such as negatively curved manifolds and trees.

What carries the argument

The one-to-one correspondence between positive invariant random fields and bounded positive harmonic functions, which carries the Martin boundary structure to the invariants.

If this is right

The collection of invariant fields is parametrized exactly by the Martin boundary.
Convergence of the deterministic heat equation to a harmonic function implies convergence of the stochastic equation to the matching invariant field.
The classification extends to any domain whose Martin boundary is nontrivial, including hyperbolic spaces and trees.
Invariant fields remain positive and bounded when the corresponding harmonic functions are.
No new invariant fields arise beyond those already present in the deterministic theory.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The result suggests that weak disorder preserves the deterministic classification rather than generating genuinely new random equilibria.
It may allow transferring known Martin boundary computations from geometry directly into statements about stochastic long-time limits.
Similar correspondences could be tested for other multiplicative equations or on graphs with different boundary structures.
Numerical approximation of invariant fields might reduce to solving the deterministic harmonic problem on the same domain.

Load-bearing premise

A natural second-moment condition holds for the multiplicative noise.

What would settle it

Finding a positive invariant field on a space with known Martin boundary, such as a regular tree, that cannot be matched to any bounded positive harmonic function.

read the original abstract

We study invariant random fields of nonlinear multiplicative stochastic heat equations in the weak disorder regime. Under a natural second-moment condition, we show that positive invariant fields are in one-to-one correspondence with bounded positive harmonic functions of the underlying space. This implies that the space of invariant fields inherits the structure of the Martin boundary. We also show that whenever the deterministic heat flow converges to a bounded harmonic function, the stochastic evolution converges to the corresponding invariant field. The results apply to many settings with nontrivial Martin boundary, such as negatively curved manifolds and trees.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives a one-to-one correspondence between positive invariant fields of the nonlinear multiplicative SHE and bounded positive harmonic functions under a second-moment condition, so the fields inherit the Martin boundary, but that condition looks unverified for the tree and manifold cases.

read the letter

The main takeaway is that positive invariant random fields for the nonlinear multiplicative stochastic heat equation in weak disorder stand in one-to-one correspondence with bounded positive harmonic functions on the underlying space. This lets the space of invariant fields carry the Martin boundary structure. The paper also shows that convergence of the deterministic heat flow to a bounded harmonic function implies convergence of the stochastic evolution to the matching invariant field. These claims are stated for settings with nontrivial Martin boundary such as negatively curved manifolds and trees. What works is the clean extension from the linear case to the nonlinear multiplicative setting and the explicit link to harmonic functions. The abstract lays out the second-moment condition and the convergence statement without obvious circularity. The soft spot is the second-moment condition itself. The paper treats it as natural, yet in geometries with exponential volume growth the Green function can grow fast enough that the second moment diverges even under weak disorder. No explicit check appears for the tree or manifold examples, so it is unclear whether the bijection and the inheritance of the Martin boundary actually hold in the regimes the abstract highlights. If the moment fails, the correspondence breaks. This paper is for people already working on stochastic PDEs in random media on non-Euclidean spaces and on invariant measures via potential theory. A reader who knows the linear theory and Martin boundaries will see the extension quickly and can judge the moment issue themselves. I would send it to peer review because the core correspondence is stated sharply enough to merit referee scrutiny on the assumptions and the examples.

Referee Report

2 major / 1 minor

Summary. The manuscript studies invariant random fields of the nonlinear multiplicative stochastic heat equation in the weak-disorder regime. Under a natural second-moment condition, it establishes a one-to-one correspondence between positive invariant fields and bounded positive harmonic functions of the underlying space, implying that the space of invariant fields inherits the structure of the Martin boundary. It further shows that convergence of the deterministic heat flow to a bounded harmonic function implies convergence of the stochastic evolution to the corresponding invariant field. The results are claimed to apply to spaces with nontrivial Martin boundary, including negatively curved manifolds and trees.

Significance. If the correspondence and convergence hold, the work provides a rigorous link between the long-time behavior of a class of nonlinear SPDEs and classical potential theory via the Martin boundary. This could be significant for analyzing invariant measures and ergodicity in random media on non-Euclidean spaces, offering a deterministic proxy for stochastic invariants in settings where direct analysis is difficult.

major comments (2)

[Applications to manifolds and trees (likely §4 or §5)] The second-moment condition is invoked as the key hypothesis for the bijection between positive invariant fields and bounded positive harmonic functions (abstract and main theorem statement). However, no explicit verification is provided that this condition holds uniformly in the weak-disorder regime for the claimed examples with nontrivial Martin boundary, such as trees (where the Green function grows exponentially) or negatively curved manifolds. If the second moment diverges due to branching or volume growth, the integrability step from the stochastic evolution to the deterministic harmonic function fails, undermining the claimed inheritance of Martin boundary structure.
[Convergence theorem (likely §3)] The convergence statement (whenever the deterministic heat flow converges to a bounded harmonic function, the stochastic evolution converges to the corresponding invariant field) relies on the same second-moment condition without error estimates or quantitative rates. This makes it difficult to assess robustness when the condition is only marginally satisfied.

minor comments (1)

[Abstract and Introduction] The abstract states the correspondence and convergence but provides no proof outline, error estimates, or verification steps; the full manuscript should include a brief roadmap of the argument in the introduction.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive assessment of our work and for the constructive major comments. We address each point below with clarifications and indicate where revisions will be made.

read point-by-point responses

Referee: The second-moment condition is invoked as the key hypothesis for the bijection between positive invariant fields and bounded positive harmonic functions. However, no explicit verification is provided that this condition holds uniformly in the weak-disorder regime for the claimed examples with nontrivial Martin boundary, such as trees or negatively curved manifolds. If the second moment diverges due to branching or volume growth, the integrability step fails.

Authors: We agree that the second-moment condition is central to the correspondence and that explicit checks for the examples strengthen the claims. The condition is stated as a hypothesis in the main theorem, but the weak-disorder regime (small noise intensity) ensures finiteness of the second moment for trees and negatively curved manifolds by controlling the exponential growth of the Green function or volume. We will add a dedicated remark in the applications section (around §4–5) with explicit bounds verifying the condition holds uniformly under the weak-disorder assumption. revision: yes
Referee: The convergence statement relies on the same second-moment condition without error estimates or quantitative rates. This makes it difficult to assess robustness when the condition is only marginally satisfied.

Authors: The convergence theorem is qualitative, using the second-moment condition to obtain almost-sure convergence via martingale arguments and uniform integrability. Quantitative rates or error estimates are not included because they would require stronger assumptions on the deterministic flow's convergence speed or higher moments, which lie outside the paper's scope. We will add a short discussion in the convergence section noting that the result remains valid whenever the second moment is finite and uniformly bounded, thereby addressing marginal satisfaction. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper derives the one-to-one correspondence between positive invariant fields and bounded positive harmonic functions from analysis of the stochastic evolution under an explicitly stated second-moment condition on the multiplicative noise. No equations, definitions, or steps in the abstract or described claims reduce this bijection to a fitted parameter, self-referential construction, or load-bearing self-citation; the Martin boundary inheritance is presented as a direct consequence rather than an input. The second-moment condition is an external assumption whose applicability is a matter of verification, not a circular reduction within the derivation chain itself. The result remains independent of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on a single domain assumption (the second-moment condition) and standard background results from stochastic analysis and potential theory; no free parameters or new entities are introduced.

axioms (1)

domain assumption natural second-moment condition on the noise
Invoked to guarantee the one-to-one correspondence between positive invariant fields and bounded positive harmonic functions.

pith-pipeline@v0.9.0 · 5372 in / 1155 out tokens · 17349 ms · 2026-05-15T21:47:11.498603+00:00 · methodology

discussion (0)

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Works this paper leans on

12 extracted references · 12 canonical work pages

[1]

Probability distribution of the free energy of the continuum directed random polymer in 1 + 1 dimensions

[ACQ10] Gideon Amir, Ivan Corwin, and Jeremy Quastel. “Probability distribution of the free energy of the continuum directed random polymer in 1 + 1 dimensions”. In:Communications on Pure and Applied Mathematics64 (2010). [ALW11] Siva Athreya, Wolfgang L¨ ohr, and Anita Winter. “Invariance principle for variable speed random walks on trees”. In:Probabilit...

work page 2010
[2]

Negatively curved manifolds, elliptic operators, and the Martin boundary

[Anc87] A. Ancona. “Negatively curved manifolds, elliptic operators, and the Martin boundary”. In: Ann. of Math.125 (1987), pp. 495–536. [AS85] M. Anderson and R. Schoen. “Positive harmonic functions on complete manifolds of negative curvature”. In:Ann. of Math.121 (1985), pp. 429–461. [Bau+23] Fabrice Baudoin, Cheng Ouyang, Samy Tindel, and Jing Wang. “P...

work page doi:10.1016/j.jfa.2023.109920 1987
[3]

Global geometry within an SPDE well-posedness prob- lem

[CO25] Hongyi Chen and Cheng Ouyang. “Global geometry within an SPDE well-posedness prob- lem”. In:Probability Theory and Related Fields(2025), pp. 1–36. [Cro08] David A. Croydon. “Heat kernel estimates for stochastic processes on resistance forms”. In: Annales de l’Institut Henri Poincar´ e, Probabilit´ es et Statistiques44.3 (2008), pp. 544–569. [CSZ20]...

work page 2025
[4]

Directed polymers in random environment are diffusive at weak disorder

[CY04] Francis Comets and Nobuo Yoshida. “Directed polymers in random environment are diffusive at weak disorder”. In:Annals of Probability34 (2004), pp. 1746–1770. [DGK23] Alexander Dunlap, Yu Gu, and Tomasz Komorowski. “Fluctuation exponents of the KPZ equation on a large torus”. In:Communications on Pure and Applied Mathematics76.11 (2023), pp. 3104–31...

work page 2004
[5]

Additive-multiplicative stochastic heat equa- tions, stationary solutions, and Cauchy statistics

Springer. 2023, pp. 2537–2555. [DM24] Alexander Dunlap and Chiranjib Mukherjee. “Additive-multiplicative stochastic heat equa- tions, stationary solutions, and Cauchy statistics”. In:The Annals of Applied Probability (2024).url:https://api.semanticscholar.org/CorpusID:267412649. [DS24] Alexander Dunlap and Evan Sorensen. “Viscous shock fluctuations in KPZ...

work page 2023
[6]

Fluctuations of a nonlinear stochastic heat equation in dimensions three and higher

Institut Henri Poincar´ e. 2024, pp. 1570–1618. [GL20] Yu Gu and Jiawei Li. “Fluctuations of a nonlinear stochastic heat equation in dimensions three and higher”. In:SIAM Journal on Mathematical Analysis52.6 (2020), pp. 5422–5440. [GO25] Xi Geng and Cheng Ouyang.Parabolic Anderson Model in Hyperbolic Spaces and Phase Transition

work page 2024
[7]

Integration by parts and invariant measure for KPZ

arXiv:2507.05530 [math.PR]. [GQ24] Yu Gu and Jeremy Quastel. “Integration by parts and invariant measure for KPZ”. In:Trans- actions of the American Mathematical Society(2024).url:https://api.semanticscholar. org/CorpusID:272654117. 13 [Gro+23] Sean Groathouse, Firas Rassoul-Agha, Timo Sepp¨ al¨ ainen, and Evan L. Sorensen. “Jointly invariant measures for...

work page arXiv 2024
[8]

Parabolic Anderson Model in the Hyperbolic Space. Part I: An- nealed Asymptotics

[GX25] Xi Geng and Weijun Xu. “Parabolic Anderson Model in the Hyperbolic Space. Part I: An- nealed Asymptotics”. In: 2025.url:https : / / api . semanticscholar . org / CorpusID : 280286878. [HN09] Yaozhong Hu and David Nualart. “Stochastic heat equation driven by fractional noise and local time”. In:Probability Theory and Related Fields143.1 (2009), pp. ...

work page doi:10.1017/s0305004118000257 2025
[9]

Replica Bethe ansatz studies of two-dimensional interfaces with quenched random impurities

arXiv:2603.18384 [math.PR].url:https://arxiv. org/abs/2603.18384. [Kar87] Mehran Kardar. “Replica Bethe ansatz studies of two-dimensional interfaces with quenched random impurities”. In:Nuclear Physics B290 (1987), pp. 582–602. [Kho14] Davar Khoshnevisan.Analysis of stochastic partial differential equations. Vol

work page arXiv 1987
[10]

Spectrum, harmonic functions, and hyperbolic metric spaces

[Kif95] Yuri Kifer. “Spectrum, harmonic functions, and hyperbolic metric spaces”. In:Israel J. Math. 89.1-3 (1995), pp. 377–428.issn: 0021-2172,1565-8511.doi:10.1007/BF02808210. [KPZ86] Mehran Kardar, Giorgio Parisi, and Yi-Cheng Zhang. “Dynamic scaling of growing inter- faces”. In:Physical Review Letters56.9 (1986), p

work page doi:10.1007/bf02808210 1995
[11]

The random uniform shear layer: an explicit example of turbulent diffusion with broad tail probability distributions

[Maj93] Andrew J Majda. “The random uniform shear layer: an explicit example of turbulent diffusion with broad tail probability distributions”. In:Physics of Fluids A: Fluid Dynamics5.8 (1993), pp. 1963–1970. [Mol91] Stanislav Molchanov. “Ideas in the theory of random media”. In:Acta Applicandae Mathe- matica22 (1991), pp. 139–282. [Nee21] Robert W. Neel....

work page doi:10.30757/alea.v18-61 1993
[12]

Martin boundaries of random walks: ends of trees and groups

[PW87] Massimo A. Picardello and Wolfgang Woess. “Martin boundaries of random walks: ends of trees and groups”. In:Transactions of the American Mathematical Society302.1 (1987), pp. 185–205. [TV02] Samy Tindel and Frederi Viens. “Almost sure exponential behaviour for a parabolic SPDE on a manifold”. In:Stochastic Processes and their Applications100.1 (200...

work page doi:10.1016/s0304-4149(02)00102-3 1987