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arxiv: 2501.07325 · v2 · pith:R2YKW5UZnew · submitted 2025-01-13 · 🧮 math.PR

Large deviation principle for the stationary solutions of stochastic functional differential equations with infinite delay

Yong Liu , Bin Tang This is my paper

Pith reviewed 2026-05-23 05:51 UTC · model grok-4.3

classification 🧮 math.PR
keywords large deviation principlestationary solutionsstochastic functional differential equationsinfinite delayinvariant measuresweak convergence approach
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The pith

Stationary solutions of SFDEs with infinite delay satisfy a large deviation principle under small noise.

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

This paper first proves existence and uniqueness of stationary solutions for stochastic functional differential equations with infinite delay. It then uses the weak convergence approach to establish a uniform large deviation principle for the solution maps, from which the LDP for stationary solutions follows directly. The LDP for invariant measures is obtained next by applying the contraction principle. A reader would care because the result quantifies the exponential cost of rare long-term behaviors in systems whose state depends on the entire infinite past.

Core claim

The authors demonstrate the existence and uniqueness of stationary solutions. By the weak convergence approach, they show the uniform large deviation principle for the solution maps, and then prove the LDP for stationary solutions. Furthermore, they obtain the LDP for invariant measures of SFDEs through the LDP for stationary solutions and the contraction principle.

What carries the argument

Weak convergence approach to uniform LDP for solution maps, followed by the contraction principle.

Load-bearing premise

Existence and uniqueness of the stationary solutions must hold before any large deviation principle can be stated for them.

What would settle it

An explicit SFDE with infinite delay that possesses unique stationary solutions yet whose deviation probabilities fail to obey the claimed rate function.

read the original abstract

We investigate the large deviation principle (LDP) of the stationary solutions of stochastic functional differential equations (SFDEs) with infinite delay under small random perturbation. First, we demonstrate the existence and uniqueness of the corresponding stationary solutions. Second, by the weak convergence approach, we show the uniform large deviation principle for the solution maps, and then prove the LDP for stationary solutions. Furthermore, we obtain the LDP for invariant measures of SFDEs through the LDP for stationary solutions and the contraction principle.

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 / 2 minor

Summary. The manuscript proves existence and uniqueness of stationary solutions for stochastic functional differential equations (SFDEs) with infinite delay, then establishes the uniform large deviation principle (LDP) for the solution maps via the weak convergence approach, derives the LDP for the stationary solutions themselves, and finally obtains the LDP for the associated invariant measures by applying the contraction principle.

Significance. If the results hold, the work extends large-deviation theory to SFDEs with infinite delay, a setting relevant to models with long-range memory in applied probability. The logical sequence (existence first, then uniform LDP on maps, then contraction) follows the standard Dupuis-Ellis framework and is internally consistent; the explicit treatment of the stationary-solution prerequisite removes an obvious circularity risk.

minor comments (2)
  1. The abstract states the results but the provided text does not include the detailed assumptions on the coefficients, the weighted sup-norm topology, or the compactness arguments needed to verify the weak-convergence step; these should be stated explicitly in §2 or §3.
  2. Notation for the infinite-delay segment (e.g., the phase space C_γ) and the small-noise parameter ε should be introduced with a single consistent definition early in the paper.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for summarizing the manuscript and for noting its potential relevance to models with long-range memory. The logical structure (existence of stationary solutions, uniform LDP on solution maps via weak convergence, LDP for stationary solutions, and contraction to invariant measures) is correctly identified. No major comments appear in the report, so we have no point-by-point revisions to propose. We remain available to address any specific concerns that may have led to the 'uncertain' recommendation.

Circularity Check

0 steps flagged

No significant circularity; derivation follows standard non-circular sequence

full rationale

The paper explicitly sequences its claims as: (1) prove existence/uniqueness of stationary solutions, (2) apply weak convergence to obtain uniform LDP on solution maps, (3) deduce LDP for stationary solutions, (4) apply contraction principle to reach LDP for invariant measures. This ordering ensures the prerequisite is established independently before use, with no self-definitional loops, no fitted parameters renamed as predictions, and no load-bearing self-citations. The approach is described as an adaptation of the Dupuis-Ellis framework to infinite-delay SFDEs, which is externally verifiable and does not reduce to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no information on free parameters, axioms, or invented entities.

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

    math.PR 2026-04 unverdicted novelty 7.0

    Stationary solutions of SPDEs with locally monotone coefficients satisfy the Freidlin-Wentzell LDP, from which the LDP for invariant measures follows by contraction, covering reaction-diffusion, Burgers, Navier-Stokes...

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