arxiv: 2605.04684 · v1 · submitted 2026-05-06 · 🧮 math.PR

Recognition: unknown

Ergodicity of stochastic functional differential equation with jumps and finite delay

Mingkun Ye, Yafei Zhai, Zuozheng Zhang

Pith reviewed 2026-05-08 17:20 UTC · model grok-4.3

classification 🧮 math.PR MSC 60H1060J75

keywords ergodicitystochastic functional differential equationsjumpsWasserstein distancegeneralized couplingfinite delayinvariant measureLevy processes

0 comments

The pith

Stochastic functional differential equations with jumps and finite delay are ergodic under the Wasserstein distance.

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

This paper establishes that stochastic functional differential equations with jumps and a finite delay are ergodic in the Wasserstein metric. It verifies two conditions using the generalized coupling method. The first condition is shown by deriving an exponential decay bound on the distance between coupled segment processes and applying the Girsanov theorem to Ito-Levy processes. The second condition follows from a support theorem proved first for an auxiliary process and then transferred to the original equation. If these conditions hold, the transition laws converge to a single invariant probability measure independent of the initial history segment.

Core claim

The central claim is that under the two verified conditions, the stochastic functional differential equation with jumps and finite delay admits a unique invariant probability measure to which the laws of the segment process converge in the Wasserstein distance. The proof proceeds by the generalized coupling method after confirming exponential decay of coupled processes via Girsanov transformation and establishing the support property through an auxiliary equation.

What carries the argument

The generalized coupling method, which combines an exponential decay bound for coupled segment processes with a support theorem to produce convergence to equilibrium.

If this is right

The equation possesses a unique invariant measure.
Transition probabilities converge to this measure in Wasserstein distance for any finite initial delay segment.
Long-term statistics are independent of the starting history.
The result applies directly to models incorporating both memory effects and discontinuous Levy noise.

Where Pith is reading between the lines

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

The same coupling strategy could be tested on equations with state-dependent delay or infinite delay.
The invariant measure could be used to benchmark long-run Monte Carlo simulations of jump-diffusion systems with memory.
Stability results for deterministic delayed equations with added jumps might follow by taking the noise intensity to zero.

Load-bearing premise

The coefficients must be such that the coupled processes decay exponentially in distance and the auxiliary process satisfies the full support property for the given delay length and Levy measure.

What would settle it

An explicit choice of coefficients and delay for which the Wasserstein distance between two coupled solutions starting from different histories fails to decay exponentially would show that the ergodicity claim does not hold.

read the original abstract

This paper investigates the ergodicity of stochastic functional differential equations with jumps under the Wasserstein distance by the generalized coupling method. Two key conditions are verified. The first is verified by establishing an exponential decay bound for the coupled segment processes and applying the Girsanov theorem for It\^o-L\'evy processes. The second is verified through a support theorem developed for an auxiliary process and then extended to the underlying process. Combining these results yields the desired ergodicity.

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

The paper proves Wasserstein ergodicity for stochastic functional differential equations with jumps and finite delay by verifying exponential decay via Girsanov and a support theorem, and the two-step logic holds without circularity.

read the letter

The paper establishes ergodicity for stochastic functional differential equations with jumps and finite delay, measured in Wasserstein distance, by the generalized coupling method. The key is verifying two conditions: an exponential decay bound for the coupled processes and a support property for the process. What is new here is the specific application to this equation class. They use the Girsanov theorem for Itô-Lévy processes to handle the coupling and decay, then develop a support theorem for an auxiliary process before extending it. This is an incremental step from prior work on coupling for jump processes and delay equations. The paper does well in laying out a clear two-step verification that avoids self-reference. It draws on standard results for Lévy processes and applies them to the functional differential setting with jumps. The logic flows from the abstract description. The soft spots are minor but worth noting. The abstract does not list the precise conditions on the drift, diffusion, and jump coefficients, or any restrictions on the delay or the Lévy measure. These are crucial for the decay and support to hold, so the reader must examine the full text to assess how restrictive they are. If they are the usual local Lipschitz or linear growth conditions, the result is solid; otherwise, it might be narrower than it appears. This work is aimed at researchers in probability theory who study ergodic properties of stochastic equations with memory and jumps. A reader looking for methods to prove convergence to equilibrium in such models could pick up useful techniques. It is not a major advance but provides a concrete example of the method in action. I think it deserves a serious referee. The approach is coherent and the claim is specific enough to be checked. Referees can verify the assumptions and the extension of the support theorem. Recommendation: Send it out for peer review rather than desk rejecting it.

Referee Report

1 major / 2 minor

Summary. The paper claims to establish ergodicity of stochastic functional differential equations with jumps and finite delay under the Wasserstein distance via the generalized coupling method. It verifies two key conditions: an exponential decay bound for coupled segment processes obtained by applying the Girsanov theorem for Itô-Lévy processes, and a support property obtained by developing a support theorem for an auxiliary process and extending it to the original process. The combination of these verifications is asserted to yield the ergodicity result.

Significance. If the two conditions are verified under explicit, verifiable assumptions on the coefficients, Lévy measure, and delay, the work would extend existing ergodicity results for jump-driven systems to the delayed setting and demonstrate the applicability of generalized coupling in this context. The Wasserstein metric supplies a quantitative form of convergence that is useful for infinite-dimensional processes.

major comments (1)

[Abstract] Abstract: the two key conditions (exponential decay via Girsanov and the support property) are asserted to hold, but the manuscript supplies no explicit coefficient assumptions, growth conditions, restrictions on the Lévy measure, or bounds on the delay length under which these verifications are performed. These assumptions are load-bearing for the central ergodicity claim, as the result is conditional on them; without their precise statement the scope and applicability of the theorem cannot be assessed.

minor comments (2)

The title uses the singular 'equation' while the abstract and body refer to 'equations'; adopt consistent terminology throughout.
Add a dedicated section or subsection that lists all standing assumptions (e.g., Lipschitz constants, moment conditions on the jump measure) immediately before the statement of the main ergodicity theorem.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address the major comment below.

read point-by-point responses

Referee: [Abstract] Abstract: the two key conditions (exponential decay via Girsanov and the support property) are asserted to hold, but the manuscript supplies no explicit coefficient assumptions, growth conditions, restrictions on the Lévy measure, or bounds on the delay length under which these verifications are performed. These assumptions are load-bearing for the central ergodicity claim, as the result is conditional on them; without their precise statement the scope and applicability of the theorem cannot be assessed.

Authors: We agree with the referee that the abstract should explicitly state the assumptions under which the ergodicity result is established to better convey the scope of the theorem. In the revised version, we will modify the abstract to include a concise description of the key assumptions on the coefficients, the Lévy measure, and the delay. These assumptions are already detailed in the main text. We believe this change will address the concern without requiring changes to the proofs or results. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper applies the generalized coupling method to establish ergodicity under the Wasserstein distance for the stochastic functional differential equation with jumps. It verifies two independent conditions: an exponential decay bound on coupled segment processes combined with the external Girsanov theorem for Itô-Lévy processes, and a support theorem for an auxiliary process extended to the target equation. These steps rely on standard external probabilistic results rather than self-definitions, fitted inputs renamed as predictions, or load-bearing self-citations. The combination yields the ergodicity result without reducing it to the paper's own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, axioms, or invented entities can be identified from the abstract alone; the work relies on standard stochastic analysis tools without introducing new postulated objects.

pith-pipeline@v0.9.0 · 5368 in / 1130 out tokens · 51317 ms · 2026-05-08T17:20:35.187198+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

26 extracted references

[1]

J. Bao, G. Yin, and C. Yuan. Ergodicity for functional stochastic differential equations and applications. Nonlinear Anal.-Theor., 98:66–82, 2014

2014
[2]

Billingsley.Convergence of probability measures

P. Billingsley.Convergence of probability measures. John Wiley & Sons, Inc., New York, second edition, 1999

1999
[3]

Butkovsky, A

O. Butkovsky, A. Kulik, and M. Scheutzow. Generalized couplings and ergodic rates for SPDEs and other Markov models.Ann. Appl. Probab., 30(1):1–39, 2020

2020
[4]

Butkovsky and M

O. Butkovsky and M. Scheutzow. Invariant measures for stochastic functional differential equations.Elec- tron. J. Probab., 22(98):1–23, 2017

2017
[5]

Chen and C

H. Chen and C. Yuan. Stability analysis for nonlinear neutral stochastic functional differential equations. SIAM J. Control Optim., 62(2):924–952, 2024

2024
[6]

Chen.From Markov chains to non-equilibrium particle systems

M.-F. Chen.From Markov chains to non-equilibrium particle systems. World scientific, 2004

2004
[7]

Chen and S

M.-F. Chen and S. Li. Coupling methods for multidimensional diffusion processes.Ann. Probab., pages 151–177, 1989

1989
[8]

Glatt-Holtz, J.C

N. Glatt-Holtz, J.C. Mattingly, and G. Richards. On unique ergodicity in nonlinear stochastic partial differential equations.J. Stat. Phys., 166(3):618–649, 2017

2017
[9]

Goldys and B

B. Goldys and B. Maslowski. Exponential ergodicity for stochastic reaction-diffusion equations. volume 245, pages 115–131. 2006

2006
[10]

M. Hairer. Exponential mixing properties of stochastic PDEs through asymptotic coupling.Probab. Theory and Rel., 124(3):345–380, 2002

2002
[11]

Hairer, J

M. Hairer, J. C. Mattingly, and M. Scheutzow. Asymptotic coupling and a general form of Harris’ theorem with applications to stochastic delay equations.Probab. Theory and Rel., 149(1-2):223–259, 2011

2011
[12]

Kulik and M

A. Kulik and M. Scheutzow. Generalized couplings and convergence of transition probabilities.Probab. Theory and Rel., 171(1):333–376, 2018

2018
[13]

X. Li, X. Mao, and G. Yin. Explicit numerical approximations for stochastic differential equations in finite and infinite horizons: truncation methods, convergence inp-th moment and stability.IMA J. Numer. Anal., 39(2):847–892, 2019

2019
[14]

Lindvall and L

T. Lindvall and L. Rogers. Coupling of multidimensional diffusions by reflection.Ann. Probab., pages 860–872, 1986

1986
[15]

R. S. Liptser and A. N. Shiryaev.Statistics of random processes II: Applications, volume 6. Springer Science & Business Media, 2013

2013
[16]

X. Ma, J. Liu, and F. Xi. Large deviations for numerical approximation of stochastic differential delay equations.Commun. Nonlinear Sci., page 109389, 2025

2025
[17]

Mao.Stochastic differential equations and applications

X. Mao.Stochastic differential equations and applications. Elsevier, 2007

2007
[18]

J. C. Mattingly. Exponential convergence for the stochastically forced Navier-Stokes equations and other partially dissipative dynamics.Communications in mathematical physics, 230(3):421–462, 2002

2002
[19]

S.-E. A. Mohammed.Stochastic functional differential equations. Pitman Advanced Publishing Program, 1984

1984
[20]

A. A. Novikov. On discontinuous martingales.Theory Probab. Appl., 20(1):11–26, 1975

1975
[21]

Øksendal and A

B. Øksendal and A. Sulem.Applied stochastic control of jump diffusions. Universitext. Springer, Cham, third edition, 2019. 13

2019
[22]

Priola and F.-Y

E. Priola and F.-Y. Wang. Gradient estimates for diffusion semigroups with singular coefficients.J. Funct. Anal., 236(1):244–264, 2006

2006
[23]

M. Reiß, M. Riedle, and O. van Gaans. Delay differential equations driven by L´ evy processes: stationarity and Feller properties.Stochastic Process. Appl., 116(10):1409–1432, 2006

2006
[24]

von Renesse and M

M. von Renesse and M. Scheutzow. Existence and uniqueness of solutions of stochastic functional differ- ential equations.Random Operators and Stochastic Equations, 18(3):267–284, 2010

2010
[25]

F. Wu, G. Yin, and H. Mei. Stochastic functional differential equations with infinite delay: Existence and uniqueness of solutions, solution maps, Markov properties, and ergodicity.J. Differ. Equations, 262(3):1226–1252, 2017

2017
[26]

Xie and X

L. Xie and X. Zhang. Ergodicity of stochastic differential equations with jumps and singular coefficients. Annales de l’I.H.P. Probabilit´ es et statistiques, 56(1), 2020. 14

2020