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arxiv: 2604.22461 · v1 · submitted 2026-04-24 · 🧮 math.PR

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Large deviation principles for the stationary solutions and invariant measures of a class of SPDE with locally monotone coefficients

Yong Liu , Bin Tang , Rangrang Zhang
Authors on Pith no claims yet

Pith reviewed 2026-05-08 10:07 UTC · model grok-4.3

classification 🧮 math.PR
keywords stochastic partial differential equationslarge deviation principlesstationary solutionsinvariant measureslocally monotone coefficientsFreidlin-Wentzell theorycontraction principle
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The pith

Stationary solutions of SPDEs with locally monotone coefficients satisfy the Freidlin-Wentzell large deviation principle, from which the principle for invariant measures follows by contraction.

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

The paper shows that stationary solutions exist and obey the Freidlin-Wentzell large deviation principle for a class of stochastic partial differential equations whose coefficients satisfy a local monotonicity condition. This large deviation principle for the stationary solutions transfers directly to the associated invariant measures through the contraction principle. By working with the stationary objects rather than pathwise solutions, the argument avoids constructing quasi-potentials and checking uniform large deviation estimates over bounded sets. The resulting framework accommodates additive noise, multiplicative noise, and transport-type noise, and it applies to concrete models such as stochastic reaction-diffusion equations, the 1D viscous Burgers equation, 2D Navier-Stokes equations, 2D magneto-hydrodynamic equations, and 3D hyper-dissipative Navier-Stokes equations.

Core claim

We establish the well-posedness of stationary solutions for a class of SPDEs with locally monotone coefficients, and prove the Freidlin-Wentzell large deviation principle (LDP) for these stationary solutions. The LDP for the associated invariant measures then follows via the contraction principle, avoiding the need to construct the quasi-potential and verify the Dembo-Zeitouni uniform LDP over bounded sets. By working directly with stationary solutions, we bypass these technical difficulties, thereby providing a more general and flexible framework that is adapted to additive noise, multiplicative noise, and transport-type noise.

What carries the argument

Direct application of the Freidlin-Wentzell large deviation principle to the stationary solutions, followed by the contraction principle to obtain the large deviation principle for the invariant measures.

If this is right

  • The large deviation principle holds for stationary solutions of stochastic reaction-diffusion equations with locally monotone coefficients.
  • The same large deviation principle applies to stationary solutions of the stochastic 1D viscous Burgers equation, 2D Navier-Stokes equations, 2D magneto-hydrodynamic equations, and 3D hyper-dissipative Navier-Stokes equations.
  • The invariant measures of these equations satisfy the large deviation principle by the contraction principle without separate quasi-potential construction.
  • The framework covers SPDEs driven by additive, multiplicative, or transport-type noise under the local monotonicity assumption.

Where Pith is reading between the lines

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

  • The direct stationary-solution route may simplify large-deviation analysis for other SPDEs whose coefficients obey similar local monotonicity but lack global Lipschitz continuity.
  • Explicit rate functions for specific models such as the Burgers equation could become more accessible once the contraction step is applied to stationary solutions.
  • The method might extend to proving moderate deviation principles or other concentration results by replacing the Freidlin-Wentzell scaling with different noise intensities.

Load-bearing premise

The coefficients of the SPDE must satisfy a local monotonicity condition together with growth and noise bounds that guarantee the existence of stationary solutions.

What would settle it

A counterexample in which the large deviation principle fails to hold for the stationary solutions of the stochastic 2D Navier-Stokes equations under the stated local monotonicity and growth assumptions would refute the general claim.

read the original abstract

We establish the well-posedness of stationary solutions for a class of SPDEs with locally monotone coefficients, and prove the Freidlin--Wentzell large deviation principle (LDP) for these stationary solutions. The LDP for the associated invariant measures then follows via the contraction principle, avoiding the need to construct the quasi-potential and verify the Dembo--Zeitouni uniform LDP over bounded sets. By working directly with stationary solutions, we bypass these technical difficulties, thereby providing a more general and flexible framework that is adapted to additive noise, multiplicative noise, and transport-type noise. As applications, our results cover a range of SPDEs, including the stochastic reaction-diffusion equations, stochastic 1D viscous Burgers equation, stochastic 2D Navier--Stokes equations, stochastic 2D magneto-hydrodynamic equations and stochastic 3D hyper-dissipative Navier--Stokes equations.

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

0 major / 3 minor

Summary. The manuscript establishes well-posedness of stationary solutions for a class of SPDEs with locally monotone coefficients under suitable growth and noise assumptions. It proves the Freidlin-Wentzell large deviation principle directly for the laws of these stationary solutions, then obtains the LDP for the associated invariant measures by applying the contraction principle to the continuous one-time marginal map. The framework is shown to cover additive, multiplicative, and transport-type noise, with applications to stochastic reaction-diffusion equations, the 1D viscous Burgers equation, 2D Navier-Stokes, 2D MHD, and 3D hyper-dissipative Navier-Stokes equations.

Significance. If the technical estimates hold, the work supplies a more flexible route to large-deviation results for infinite-dimensional systems by working directly with stationary solutions rather than constructing quasi-potentials or verifying uniform LDPs over bounded sets. This bypasses several technical obstacles that appear in earlier approaches and extends the range of admissible coefficients and noise structures, with immediate applicability to models in fluid dynamics and reaction-diffusion theory.

minor comments (3)
  1. §2 (Assumptions): the locally monotone condition is stated in a form that permits the listed applications, but the precise growth exponents needed for the tightness argument in the stationary setting should be displayed explicitly alongside the general hypotheses.
  2. §4 (LDP for stationary solutions): the identification of the rate function via the contraction principle is invoked after establishing the LDP on path space; a brief remark on why the marginal map is continuous in the topology used for the LDP would improve readability.
  3. §5 (Applications): the rate functions for the concrete examples (e.g., 2D Navier-Stokes) are left implicit; writing the explicit variational formula for at least one case would help readers verify the result.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the positive assessment of its significance. The report recommends minor revision but does not list any specific major comments. We are therefore in a position to address only the overall evaluation and remain available for any additional points the referee or editor may wish to raise.

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper first establishes well-posedness of stationary solutions for the SPDEs under the locally monotone coefficient, growth, and noise assumptions via a priori estimates and tightness. It then directly proves the Freidlin-Wentzell LDP for the laws of these stationary solutions. The LDP for invariant measures is obtained by applying the standard contraction principle to the continuous one-time marginal map. No step reduces by construction to a fitted parameter, self-definition, or load-bearing self-citation; the contraction principle is an external theorem, and all central arguments rely on independent estimates and identifications rather than renaming or smuggling inputs as outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on the locally monotone coefficient assumption and standard well-posedness conditions for SPDEs; no numerical fitting or new postulated entities are introduced.

axioms (2)
  • domain assumption Coefficients of the SPDE are locally monotone
    This is the defining condition for the class of equations treated in the title and abstract.
  • domain assumption Standard growth and coercivity conditions hold to guarantee existence of stationary solutions
    The paper states it establishes well-posedness, so these background conditions from SPDE theory are invoked.

pith-pipeline@v0.9.0 · 5458 in / 1406 out tokens · 67537 ms · 2026-05-08T10:07:25.571017+00:00 · methodology

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