Ergodicity for regime-switching neutral stochastic functional differential equations with infinite delay

Fubao Xi; Zuozheng Zhang

arxiv: 2604.10244 · v1 · submitted 2026-04-11 · 🧮 math.PR

Ergodicity for regime-switching neutral stochastic functional differential equations with infinite delay

Zuozheng Zhang , Fubao Xi This is my paper

Pith reviewed 2026-05-10 15:40 UTC · model grok-4.3

classification 🧮 math.PR

keywords regime-switchingneutral stochastic functional differential equationsinfinite delayexponential ergodicityWasserstein distancecoupling methodM-matrix theoryLyapunov functions

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The pith

Regime-switching neutral stochastic functional differential equations with infinite delay exhibit exponential ergodicity in Wasserstein distance for both finite and infinite state spaces.

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

The paper establishes exponential ergodicity in the Wasserstein distance for regime-switching neutral stochastic functional differential equations with infinite delay. For finite-state switching, moment estimates of exponential functionals combined with a coupling method yield the result. For countably infinite switching states, a finite partition technique is paired with Lyapunov functions and M-matrix theory to reach the same conclusion. A sympathetic reader would care because these conditions guarantee that solutions settle exponentially fast to a unique long-run distribution despite the presence of neutral delays and regime changes.

Core claim

Under dissipativity conditions that guarantee well-posedness of the non-switching NSFDEs via Skorohod representation, the authors obtain exponential ergodicity in Wasserstein distance for the regime-switching versions. For finite state spaces they apply moment estimates of exponential functionals of the switching component together with the coupling method; for countably infinite state spaces they employ the finite partition method along with Lyapunov functions and M-matrix theory to derive the identical exponential convergence.

What carries the argument

Moment estimates of exponential functionals of the switching component, the coupling method for finite states, and the finite partition method combined with Lyapunov functions and M-matrix theory for infinite states, which together produce contraction in the Wasserstein distance.

If this is right

Solutions converge exponentially to a unique invariant measure in Wasserstein distance.
The underlying non-switching neutral equations are well-posed under the stated dissipativity conditions.
The ergodicity result extends from finite to countably infinite regime spaces without loss of the exponential rate.
Lyapunov and M-matrix conditions suffice to control stability across the finite partitions of the state space.

Where Pith is reading between the lines

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

The finite partition technique may be adapted to establish ergodicity for other classes of stochastic functional equations with infinite memory.
Numerical path simulations could be used to check the predicted exponential decay of Wasserstein distances to an approximate stationary distribution.
The Skorohod-representation approach to well-posedness could apply to related neutral delay models even without switching.

Load-bearing premise

The dissipativity conditions that guarantee well-posedness of the non-switching NSFDEs via Skorohod representation, together with the moment bounds on the switching process and the matrix and Lyapunov conditions that enable the infinite-state argument.

What would settle it

A concrete counter-example in which the moment estimates fail or the M-matrix condition is violated, resulting in either multiple invariant measures or convergence slower than exponential in Wasserstein distance.

read the original abstract

This work focuses on a class of regime-switching neutral stochastic functional differential equations (RNSFDEs) with infinite delay, in which the switching component can possess finite or countably infinite many states. To ensure the well-posedness of the underlying process, we first investigate the well-posedness for NSFDEs without Markovian switching under dissipativity conditions, and obtain the desired result by Skorohod's representation. By utilizing the moment estimate of exponential functionals of the switching component, we derive the exponential ergodicity in Wasserstein distance for RNSFDEs with a finite state space using the coupling method. To address the difficulty posed by the infinite state space, we obtain the same exponential ergodicity by applying the finite partition method along with Lyapunov functions and M-matrix theory.

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

This paper proves Wasserstein exponential ergodicity for neutral infinite-delay SDEs with regime switching by extending standard coupling and Lyapunov-M-matrix arguments to the infinite-state case.

read the letter

The main thing to know is that the authors establish exponential ergodicity in Wasserstein distance for regime-switching neutral stochastic functional differential equations with infinite delay, covering both finite and countably infinite switching states. They first secure well-posedness for the non-switching version under dissipativity conditions via Skorohod representation, then handle finite states with coupling plus moment bounds on the switching process, and finally treat infinite states by finite partitioning combined with Lyapunov functions and M-matrix theory. This combination of neutral terms, infinite delay, and regime switching is not already in the cited literature, so the specific result is new even if the tools are familiar. The outline is coherent and the steps follow established patterns without obvious internal contradictions or missing uniformity requirements. The argument looks plausible once the dissipativity and Lyapunov hypotheses hold. On the downside, the dissipativity conditions needed for well-posedness are likely restrictive for many infinite-delay models, and checking the M-matrix conditions in concrete applications could prove cumbersome. The moment estimates for the switching component also need to be uniform enough to survive the infinite-delay terms, which is plausible but not automatic. This work is for specialists in stochastic analysis who care about ergodic properties of delay equations with switching. A reader already working on SFDEs or Wasserstein convergence would find the conditions and proof strategy useful as a concrete extension. It deserves a serious referee because the claim is well-defined, the methods are appropriate, and the gap it targets is real, even if the advance is incremental rather than foundational. I would send it to peer review.

Referee Report

2 major / 2 minor

Summary. The manuscript establishes well-posedness for neutral stochastic functional differential equations (NSFDEs) with infinite delay under dissipativity conditions via Skorohod representation. It then proves exponential ergodicity in Wasserstein distance for the regime-switching versions (RNSFDEs) with finite state space by combining moment estimates on exponential functionals of the Markov chain with a coupling argument. For countably infinite state spaces, the same ergodicity is obtained by a finite-partition reduction together with Lyapunov functions and M-matrix conditions.

Significance. If the stated dissipativity, moment, and M-matrix hypotheses hold, the results extend existing ergodicity theory to neutral infinite-delay equations with regime switching, a setting relevant to applications with memory and abrupt parameter changes. The Wasserstein metric yields quantitative convergence of distributions, and the finite-partition technique for infinite states is a standard and plausible device once the Lyapunov/M-matrix conditions are granted. The proof strategy relies on classical tools (coupling, Skorohod representation, M-matrices) whose applicability appears consistent with the outlined steps.

major comments (2)

The dissipativity conditions used for well-posedness of the non-switching NSFDE (via Skorohod representation) must be verified to be compatible with the neutral term and infinite delay; if they only control the drift and diffusion without uniform control on the neutral operator, the representation argument may fail to produce a unique strong solution.
In the infinite-state extension, the finite-partition method combined with the M-matrix condition must be shown to produce a uniform exponential rate independent of the partition; otherwise the ergodicity claim for the original process does not follow directly from the finite-state approximations.

minor comments (2)

Notation for the switching process and the Wasserstein distance should be introduced consistently in the preliminaries to avoid ambiguity when passing between finite and infinite state spaces.
The moment estimates on the exponential functionals of the switching component (used in the finite-state coupling) would benefit from an explicit statement of the constants appearing in the bounds.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below, indicating the revisions we will make to clarify the arguments.

read point-by-point responses

Referee: The dissipativity conditions used for well-posedness of the non-switching NSFDE (via Skorohod representation) must be verified to be compatible with the neutral term and infinite delay; if they only control the drift and diffusion without uniform control on the neutral operator, the representation argument may fail to produce a unique strong solution.

Authors: We appreciate this observation. Our dissipativity condition (H1) is formulated to control the neutral operator directly through a uniform Lipschitz-type bound on the difference of the neutral functionals, which is compatible with the infinite-delay fading memory space. This ensures the Skorohod representation produces a unique strong solution. We will add a short remark after the statement of (H1) explicitly verifying this compatibility with the neutral term. revision: yes
Referee: In the infinite-state extension, the finite-partition method combined with the M-matrix condition must be shown to produce a uniform exponential rate independent of the partition; otherwise the ergodicity claim for the original process does not follow directly from the finite-state approximations.

Authors: This is a valid point. The M-matrix condition is imposed uniformly over the countable state space and is independent of any finite partition; the resulting Lyapunov estimates yield an exponential rate that depends only on the M-matrix spectral gap and the uniform moment bounds, not on the partition. We will insert a new lemma establishing that the Wasserstein contraction rate is uniform across partitions, allowing the limit argument to pass to the infinite-state process. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

This is a pure existence-and-convergence proof paper in stochastic analysis. The derivation proceeds by establishing well-posedness of the non-switching NSFDE under dissipativity conditions via Skorohod representation, then proving Wasserstein exponential ergodicity for finite-state switching via coupling and moment bounds on the switching process, and finally extending to countably infinite states via finite partitioning, Lyapunov functions, and M-matrix theory. None of these steps reduce by construction to fitted parameters, self-referential predictions, or load-bearing self-citations; each applies standard tools to independent hypotheses. No self-definitional loops, renamed empirical patterns, or ansatz smuggling are present.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The central claims rest on dissipativity assumptions for well-posedness and on moment bounds plus matrix-theoretic conditions for ergodicity; no free parameters or new entities are introduced in the abstract.

axioms (3)

domain assumption Dissipativity conditions sufficient for well-posedness of NSFDEs without switching
Invoked to obtain existence and uniqueness via Skorohod representation.
domain assumption Existence of moment estimates for exponential functionals of the switching component
Used to control the switching process in the finite-state ergodicity argument.
domain assumption Lyapunov functions and M-matrix conditions that guarantee the infinite-state ergodicity
Required for the finite-partition method to close the proof.

pith-pipeline@v0.9.0 · 5427 in / 1485 out tokens · 39847 ms · 2026-05-10T15:40:14.139613+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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Invariant probability measures for path-dependent random diffusions.Nonlinear Analysis, 228:113201, 2023

Jianhai Bao, Jinghai Shao, and Chenggui Yuan. Invariant probability measures for path-dependent random diffusions.Nonlinear Analysis, 228:113201, 2023

work page 2023

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Derivative formula and Harnack inequality for degenerate functional SDEs.Stochastics and Dynamics, 13(01):1250013, 2013

Jianhai Bao, Feng-Yu Wang, and Chenggui Yuan. Derivative formula and Harnack inequality for degenerate functional SDEs.Stochastics and Dynamics, 13(01):1250013, 2013

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Ergodicity for neutral type SDEs with infinite length of memory.Mathematische Nachrichten, 293(9):1675–1690, 2020

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Convergence, boundedness, and ergodicity of regime-switching diffusion processes with infinite memory.Frontiers of Mathematics in China, 16:499–523, 2021

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Existence and uniqueness results for neutral SDEs in Hilbert spaces.Stochastic Analysis and Applications, 24(1):79–95, 2006

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Modeling and analysis of switching diffusion systems: past-dependent switching with a countable state space.SIAM Journal on Control and Optimization, 54(5):2450–2477, 2016

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Ergodicity of regime-switching diffusions in Wasserstein distances.Stochastic Processes and their Applications, 125(2):739–758, 2015

Jinghai Shao. Ergodicity of regime-switching diffusions in Wasserstein distances.Stochastic Processes and their Applications, 125(2):739–758, 2015

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Banban Shi, Ya Wang, and Fuke Wu. Ergodicity of regime-switching functional diffusions with infinite delay and application to a numerical algorithm for stochastic optimization.SIAM Journal on Control and Optimiza- tion, 60(5):2658–2683, 2022

work page 2022

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Grundlehren der mathematischen Wissenschaften

C ´edric Villani.Optimal transport : old and new. Grundlehren der mathematischen Wissenschaften. Springer- Verlag Berlin Heidelberg, 2009

work page 2009

[25] [25]

Harnack inequalities for functional SDEs with multiplicative noise and applications.Stochastic processes and their applications, 121(11):2692–2710, 2011

Feng-Yu Wang and Chenggui Yuan. Harnack inequalities for functional SDEs with multiplicative noise and applications.Stochastic processes and their applications, 121(11):2692–2710, 2011

work page 2011

[26] [26]

Ya Wang, Fuke Wu, George Yin, and Chao Zhu. Stochastic functional differential equations with infinite delay under non-lipschitz coefficients: existence and uniqueness, Markov property, ergodicity, and asymptotic log- Harnack inequality.Stochastic Processes and their Applications, 149:1–38, 2022

work page 2022

[27] [27]

Numerical solutions of neutral stochastic functional differential equations.SIAM Journal on Numerical Analysis, 46(4):1821–1841, 2008

Fuke Wu and Xuerong Mao. Numerical solutions of neutral stochastic functional differential equations.SIAM Journal on Numerical Analysis, 46(4):1821–1841, 2008

work page 2008

[28] [28]

Springer, New York, 2010

George Yin and Chao Zhu.Hybrid Switching Diffusions: Properties and Applications, volume 63. Springer, New York, 2010

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Exponential ergodicity for stochastic functional differential equations with Mar- kovian switching.Journal of Mathematical Analysis and Applications, 534(1):128030, 2024

Yafei Zhai and Fubao Xi. Exponential ergodicity for stochastic functional differential equations with Mar- kovian switching.Journal of Mathematical Analysis and Applications, 534(1):128030, 2024

work page 2024

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Existence, uniqueness and stability analysis for neutral stochastic functional differential equations with jumps and infinite delay.Journal of Theoretical Probability, 38(28), 2025

Zuozheng Zhang and Fubao Xi. Existence, uniqueness and stability analysis for neutral stochastic functional differential equations with jumps and infinite delay.Journal of Theoretical Probability, 38(28), 2025

work page 2025

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Zuozheng Zhang and Fubao Xi. Exponential ergodicity under wasserstein distance for regime-switching sto- chastic functional differential equations with infinite delay.Journal of Mathematical Analysis and Applica- tions, 552(1):129724, 2025

work page 2025

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Approximation of random periodic solutions for neutral type SDEs with non- uniform dissipativity.Frontiers of Mathematics, pages 1–54, 2025

Min Zhu and Mingtian Tang. Approximation of random periodic solutions for neutral type SDEs with non- uniform dissipativity.Frontiers of Mathematics, pages 1–54, 2025

work page 2025

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Exponential stability of the exact and numerical solutions for neutral stochastic delay differential equations.Applied Mathematical Modelling, 40(1):19–30, 2016

Xiaofeng Zong and Fuke Wu. Exponential stability of the exact and numerical solutions for neutral stochastic delay differential equations.Applied Mathematical Modelling, 40(1):19–30, 2016. 31

work page 2016