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arxiv: 2607.01611 · v1 · pith:LZP5PKXKnew · submitted 2026-07-02 · 🧮 math.AP · math.PR

Invariant Measure of the Camassa-Holm Equation with Linear Multiplicative Noise

Wei Luo , Zhaoyang Yin , Pei Zheng This is my paper

Pith reviewed 2026-07-03 10:14 UTC · model grok-4.3

classification 🧮 math.AP math.PR
keywords Camassa-Holm equationlinear multiplicative noiseinvariant measurecontinuous dependencestochastic PDEshallow water model
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The pith

The solution map of the Camassa-Holm equation with linear multiplicative noise depends almost surely continuously on initial data in H^s for s>3/2, and the equation admits a non-unique invariant measure.

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

This paper shows that solutions to the Camassa-Holm equation driven by linear multiplicative noise vary continuously with the starting data, almost surely, when the data lies in a Sobolev space H^s with s exceeding 3/2. The continuous dependence property is then used to establish the existence of an invariant measure for the stochastic equation. The measure is shown to be non-unique. These results describe the well-posedness and long-time statistical properties of a stochastic model for shallow water waves.

Core claim

The authors prove that the solution map depends almost surely continuously on the deterministic initial data in H^s for s>3/2. They further prove the existence and non-uniqueness of an invariant measure for this stochastic Camassa-Holm equation.

What carries the argument

The almost sure continuous dependence of the solution map on initial data in the space H^s with s>3/2, which enables construction of the invariant measure.

Load-bearing premise

The threshold s>3/2 is high enough for all estimates controlling the nonlinear terms, the nonlocal projection P[u], and the multiplicative noise to close without losing regularity.

What would settle it

Two sequences of initial data converging in H^{3/2} whose corresponding solutions diverge with positive probability would disprove the continuous dependence claim.

read the original abstract

In this paper, we prove that the solution map of Camassa-Holm equation with linear multiplicative noise $$ \left\{ \begin{array}{l} {\rm d}u+(u\partial_xu+\partial_xP[u])\,{\rm d}t=\beta u\,{\rm d}W, u(0,x)=u_0(x), P[u]=(1-\partial_x^2)^{-1}\left(u^2+\frac 1 2(\partial_x u)^2\right) \end{array} \right. $$ depends almost surely continuously on the deterministic initial data in $H^s$ for $s>3/2$. Furthermore, we prove the existence and non-uniqueness of an invariant measure for the Camassa-Holm equation with linear multiplicative noise.

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.

Referee Report

2 major / 1 minor

Summary. The paper claims to prove that the solution map of the Camassa-Holm equation with linear multiplicative noise depends almost surely continuously on deterministic initial data in H^s for s>3/2, and furthermore establishes the existence and non-uniqueness of an invariant measure.

Significance. If the almost-sure continuous dependence of the solution map holds with a single u0-independent null set, the result would enable a well-defined Markov semigroup on H^s and thereby support the construction of invariant measures for this stochastic Camassa-Holm equation, extending known deterministic well-posedness results to the multiplicative-noise setting.

major comments (2)
  1. [continuous dependence argument] The section establishing almost-sure continuous dependence of the solution map (the argument following the integral formulation of the SPDE): the claim requires a single set Ω_0 with P(Ω_0)=1 such that u0 ↦ u(·,ω,u0) is continuous H^s → C([0,T];H^s) for every ω ∈ Ω_0. Standard difference estimates via Gronwall on the integral equation for u-v produce, for each fixed pair u0,v0, a null set N_{u0,v0} outside which the estimate holds; because the random constants depend on the trajectories (via the nonlinear terms u∂_x u, P[u] and the multiplicative noise βu dW), these null sets may depend on u0. Without an explicit uniform construction (e.g., a single Borel-Cantelli argument or u0-independent bound on the stochastic integral), the exceptional set can depend on the initial datum, so the map fails to be well-defined as a continuous function of u0 for a fixed ω. This directly undermines both
  2. [invariant measure construction] The construction of the invariant measure (the section following the continuous-dependence result): the existence and non-uniqueness statements rely on the Markov property of the solution map, which in turn requires the almost-sure continuity to hold with a u0-independent null set. If the null-set issue is not resolved, the semigroup is not well-defined on the full space and the Krylov-Bogoliubov or tightness arguments cannot be applied directly.
minor comments (1)
  1. [abstract and introduction] The abstract states the claims but supplies no proof outline, error estimates, or method description; the introduction should include a brief sketch of the key estimates and the strategy for obtaining a u0-independent null set.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and insightful comments. The concerns about ensuring a u0-independent null set for almost-sure continuous dependence, and the consequent implications for the Markov semigroup and invariant measures, are well-taken. We address each point below and will revise the manuscript accordingly to make the arguments fully rigorous.

read point-by-point responses

  1. Referee: [continuous dependence argument] The section establishing almost-sure continuous dependence of the solution map (the argument following the integral formulation of the SPDE): the claim requires a single set Ω_0 with P(Ω_0)=1 such that u0 ↦ u(·,ω,u0) is continuous H^s → C([0,T];H^s) for every ω ∈ Ω_0. Standard difference estimates via Gronwall on the integral equation for u-v produce, for each fixed pair u0,v0, a null set N_{u0,v0} outside which the estimate holds; because the random constants depend on the trajectories (via the nonlinear terms u∂_x u, P[u] and the multiplicative noise βu dW), these null sets may depend on u0. Without an explicit uniform construction (e.g., a single Borel-Cantelli argument or u0-independent bound on the stochastic integral), the exceptional set can depend on the initial datum, so the map fails to be well-defined as a continuous function of u0 for a fixed

    Authors: We agree that the current write-up of the continuous dependence does not explicitly construct a u0-independent null set and that standard per-pair Gronwall estimates alone are insufficient. In the revision we will add the following explicit argument: since H^s is separable, fix a countable dense subset {u0^k} of initial data. For each pair, the difference estimate yields a null set N_k; the union over the countable collection remains null. On the complement Ω_0, the solution map is continuous at each u0^k. For general u0 we then pass to the limit using a priori bounds on solutions that are uniform on bounded sets in H^s; these bounds follow from the linear multiplicative structure of the noise, which permits application of the Burkholder–Davis–Gundy inequality with constants depending only on the H^s-norm of the initial datum (independent of the particular trajectory beyond that norm). The resulting single null set Ω_0 works for all u0 simultaneously. revision: yes

  2. Referee: [invariant measure construction] The construction of the invariant measure (the section following the continuous-dependence result): the existence and non-uniqueness statements rely on the Markov property of the solution map, which in turn requires the almost-sure continuity to hold with a u0-independent null set. If the null-set issue is not resolved, the semigroup is not well-defined on the full space and the Krylov-Bogoliubov or tightness arguments cannot be applied directly.

    Authors: With the revised continuous-dependence argument establishing a single Ω_0, the solution map is a well-defined continuous random dynamical system for almost every ω. This yields a Markov semigroup on the full space H^s. The existence of an invariant measure then follows from the standard Krylov–Bogoliubov procedure once tightness of the time-averaged measures is verified (which relies only on the uniform moment bounds already present in the manuscript). Non-uniqueness is obtained by exhibiting two distinct stationary measures supported on different classes of solutions; this construction is independent of the null-set issue once the semigroup is well-defined. The revision to the first section therefore resolves the second concern as well. revision: partial

Circularity Check

0 steps flagged

No circularity: existence proof for invariant measure is self-contained

full rationale

The paper presents a mathematical existence proof for almost-sure continuous dependence of the solution map and for an invariant measure of the stochastic Camassa-Holm equation. No steps reduce by definition or construction to fitted parameters, self-citations that bear the central load, or ansatzes smuggled from prior work by the same authors. The derivation relies on standard stochastic PDE estimates (Gronwall, Itô formula, tightness) applied to the given integral equation; these are independent of the target result. The reader's circularity score of 1.0 is consistent with this assessment. The skeptic concern about u0-dependent null sets is a potential gap in the argument, not a circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard functional-analytic and stochastic-calculus background for SPDEs; no free parameters or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption Sobolev embedding and product estimates hold in H^s for s > 3/2
    Invoked to control the nonlinear terms and the noise in the continuous-dependence statement.
  • domain assumption The linear multiplicative noise term defines a well-posed stochastic integral in the chosen function space
    Required for the SPDE to be meaningful.

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