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arxiv: 2606.27324 · v1 · pith:54FPNNMNnew · submitted 2026-06-25 · 🧮 math.PR

Kolmogorov equations for stochastic convective Brinkman-Forchheimer equations forced by L\'evy Noise and its application to infinite horizon problems

Sagar Gautam , Manil T. Mohan This is my paper

Pith reviewed 2026-06-26 02:38 UTC · model grok-4.3

classification 🧮 math.PR
keywords Kolmogorov equationm-dissipativityLévy noiseBrinkman-Forchheimer equationsstochastic optimal controlHamilton-Jacobi-Bellman equation
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The pith

The Kolmogorov operator for Lévy-driven convective Brinkman-Forchheimer equations is essentially m-dissipative due to the absorption term.

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

This paper establishes essential m-dissipativity of the Kolmogorov operator for the stochastic convective Brinkman-Forchheimer equations with Lévy noise in two and three dimensions. It does so by relying on the structure of the absorption term rather than exponential moment estimates. The result is applied to prove solvability of the infinite-dimensional Hamilton-Jacobi-Bellman equation for an infinite-horizon stochastic optimal control problem. A sympathetic reader cares because the property supports well-posedness for the associated Markov process and control problems without strong auxiliary estimates.

Core claim

The paper claims that the Kolmogorov operator corresponding to the stochastic two- and three-dimensional incompressible convective Brinkman-Forchheimer equations driven by Lévy noise is essentially m-dissipative. This follows from exploiting the dissipativity generated by the absorption term αu + β|u|^{r-1}u with β > 0 and r sufficiently large. The same framework yields solvability of the associated infinite-dimensional Hamilton-Jacobi-Bellman integro-differential equation for an infinite-horizon stochastic optimal control problem.

What carries the argument

The absorption term αu + β|u|^{r-1}u, which supplies the dissipativity that establishes essential m-dissipativity of the Kolmogorov operator without exponential moments.

If this is right

  • Essential m-dissipativity holds for the Kolmogorov operator in both two and three dimensions on the torus.
  • The result directly yields solvability of the infinite-dimensional Hamilton-Jacobi-Bellman equation.
  • The approach applies to infinite-horizon stochastic optimal control problems under Lévy noise.
  • The framework covers the given form of multiplicative and additive jump noise.

Where Pith is reading between the lines

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

  • The method may extend to other fluid equations that contain comparable nonlinear damping.
  • It could simplify proofs of invariant measures for additional classes of jump-driven stochastic PDEs.
  • Similar structural arguments might apply to finite-horizon control problems with the same noise.

Load-bearing premise

The absorption term must take the specific form αu + β|u|^{r-1}u with β > 0 and r large enough to generate dissipativity.

What would settle it

A concrete calculation showing that when the absorption term is removed or weakened the Kolmogorov operator fails to be essentially m-dissipative.

read the original abstract

This article examines the Kolmogorov equation corresponding to the following stochastic two- and three-dimensional incompressible ($\nabla\cdot\boldsymbol{u}=0$) convective Brinkman-Forchheimer equations, also known as the damped Navier-Stokes equations, driven by L\'evy noise on the torus: \begin{align*} \mathrm{d}\boldsymbol{u}+[-\mu\Delta\boldsymbol{u}+(\boldsymbol{u}\cdot\nabla)\boldsymbol{u}+\alpha\boldsymbol{u}+\beta|\boldsymbol{u}|^{r-1}\boldsymbol{u}+\nabla p]\mathrm{d} t =\sqrt{\mathrm{Q}}\mathrm{d}\mathrm{W}+\int_{Z}\sigma(t,z)\widetilde{\pi}(\mathrm{d} t,\mathrm{d} z), \end{align*} where $\mu,\alpha,\beta>0$ are physical constants; $\mathrm{Q}$ is a non-negative, trace-class operator; $\mathrm{W}$ is a cylindrical Wiener process on a Hilbert space; $\sigma$ represents the jump-noise coefficient; $(Z,\mathscr{B}(Z))$ is a measurable space; $\pi$ is a time-homogeneous Poisson random measure; and $\widetilde{\pi}$ denotes its compensator. The main contribution of this work is the establishment of the essential $m$-dissipativity of the corresponding Kolmogorov operator, a property that has received limited attention in the existing literature for systems driven by jump-type noise. \emph{Our main innovation is that, in contrast to traditional techniques which crucially depend on exponential moment estimates, we utilize the intrinsic structure of the absorption term $\alpha\boldsymbol{u}+\beta|\boldsymbol{u}|^{r-1}\boldsymbol{u}$ to dispense with these requirements. This allows us to establish the essential $m$-dissipativity of the Kolmogorov operator without the need for exponential moments.} We apply the developed framework to an infinite-horizon stochastic optimal control problem, demonstrating the solvability of the associated infinite-dimensional Hamilton-Jacobi-Bellman (integro-differential) equation.

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

1 major / 2 minor

Summary. The manuscript studies the Kolmogorov equation associated to the stochastic convective Brinkman-Forchheimer (damped Navier-Stokes) system on the torus in dimensions 2 and 3, driven by additive Lévy noise. The central claim is that the Kolmogorov operator is essentially m-dissipative, proved by exploiting the structure of the absorption term αu + β|u|^{r-1}u (with β>0 and r sufficiently large) rather than exponential moment estimates; the result is then used to obtain existence for the associated infinite-horizon stochastic optimal control problem via the infinite-dimensional HJB integro-differential equation.

Significance. If the m-dissipativity result is established, the work contributes to the theory of Kolmogorov operators for SPDEs with jump noise by removing a common technical requirement (exponential moments). The structural use of the nonlinear damping term is a standard device in damped fluid equations and appears well-suited to the Lévy setting; the control application illustrates a concrete use case. The overall significance is moderate and depends on the generality of the assumptions retained on the Lévy measure and the precise range of r.

major comments (1)
  1. [Abstract / Main theorem on essential m-dissipativity] The abstract asserts that the absorption term dispenses with exponential moment estimates, yet the precise lower bound on r (and any dimension-dependent restrictions) is not stated; this bound is load-bearing for the dissipativity argument and must appear explicitly in the statement of the main theorem on m-dissipativity.
minor comments (2)
  1. [Preliminaries] Notation for the compensated Poisson measure and the jump coefficient σ should be introduced once in a preliminary section and used consistently thereafter.
  2. [Introduction] The abstract mentions both 2D and 3D cases; any differences in the admissible range of r or in the proof strategy between dimensions should be highlighted in the introduction.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and the recommendation of minor revision. We address the single major comment below.

read point-by-point responses

  1. Referee: [Abstract / Main theorem on essential m-dissipativity] The abstract asserts that the absorption term dispenses with exponential moment estimates, yet the precise lower bound on r (and any dimension-dependent restrictions) is not stated; this bound is load-bearing for the dissipativity argument and must appear explicitly in the statement of the main theorem on m-dissipativity.

    Authors: We agree that the explicit lower bound on r (together with any dimension-dependent restrictions) must be stated in the main theorem on essential m-dissipativity, as this threshold is essential to the dissipativity argument. The present version refers only to “r sufficiently large.” In the revised manuscript we will insert the precise condition on r directly into the statement of the main theorem and update the abstract to match. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper establishes essential m-dissipativity of the Kolmogorov operator via a direct structural argument that exploits the specific form of the absorption term αu + β|u|^{r-1}u (with β > 0 and r large) to avoid exponential moment estimates. This is a standard technique in damped Navier-Stokes analysis and does not reduce to any fitted parameter, self-referential definition, or load-bearing self-citation chain. The derivation is presented as self-contained and independent of the target result, with no steps that equate a prediction or uniqueness claim to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim rests on standard properties of Lévy noise and Poisson measures plus the specific dissipative structure of the absorption term; no new entities are introduced and no parameters are fitted inside the proof.

axioms (2)
  • domain assumption The noise is driven by a cylindrical Wiener process and a compensated Poisson random measure with the given intensity.
    Invoked in the statement of the SPDE (abstract equation).
  • domain assumption The absorption term αu + β|u|^{r-1}u with β>0 supplies sufficient dissipativity to close the estimates.
    Central to the innovation paragraph; if false the avoidance of exponential moments fails.

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