Existence and vanishing noise limit of measure attractors for McKean-Vlasov $p$-Laplacian lattice systems with delay driven by L\'evy noise

Bixiang Wang; Xiaoxiao Sun; Zhang Chen

REVIEW 3 major objections 4 minor 63 references

Existence and vanishing noise limit of measure attractors for McKean-Vlasov $p$-Laplacian lattice systems with delay driven by L\'evy noise

T0 review · 3 major / 4 minor · reviewed 2026-08-02 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read The laws of a delayed McKean-Vlasov p-Laplacian lattice system with Lévy noise admit a unique pullback measure attractor, and this attractor approaches the deterministic one at the optimal √ε rate.

desk verdict First measure-attractor theory for McKean-Vlasov delay lattice SPDEs with superlinear Lévy noise; the main gap is imported well-posedness for the frozen equation. read the letter →

arxiv 2607.11773 v2 pith:MXTVWUKD submitted 2026-07-13 math.PR

Zhang Chen , Xiaoxiao Sun , Bixiang Wang This is my paper · ORCID

classification math.PR MSC 37L5537L3037L6060G51

keywords McKean-Vlasovequationp-LaplacianlatticesystemLévynoisetimedelaySkorohodspacemeasureattractorpullbackconvergencerate

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper targets a non-autonomous McKean-Vlasov stochastic p-Laplace lattice system with time delay, driven by Brownian and Lévy noise whose coefficients may grow superlinearly. The central claim is that, for every sufficiently small noise intensity ε, the laws of the delayed segment process form a unique non-autonomous family of probability measures — a pullback measure attractor — that is invariant and attracts all bounded initial-law families; and that as ε→0 this family converges to the deterministic pullback family at the sharp rate √ε in Wasserstein distance of order θ. Two obstacles shape the proof: the Skorohod topology on càdlàg paths makes the law-evolution map continuous only for times at least as large as the delay ρ, so invariance must be recovered through a two-step cocycle argument; and the superlinear Lévy noise rules out classical moment-based equicontinuity, so pullback compactness is established through a Skorohod tightness criterion on truncated segments. If correct, the results give the first long-time statistical description, and the optimal small-noise asymptotics, for this class of mean-field delay lattice systems.

What carries the argument

The load-bearing device is the non-autonomous cocycle Φ(t,τ) on the space P_θ(D_ρ) of probability laws with finite θ-th moment on the Skorohod space D([-ρ,0], ℓ²), where ℓ² is the chain's state space; the cocycle pushes initial laws forward to laws of the delayed segment. Existence and uniqueness of solutions come from a contraction fixed point on the distribution-frozen auxiliary equation obtained by freezing the law µ in all coefficients. Pullback absorbing sets come from weighted exponential moment estimates; pullback asymptotic compactness comes from a Skorohod tightness criterion applied to finite-dimensional truncations of the segment process, bypassing the unavailable equicontinuity f

What would settle it

Check whether the cited well-posedness theorem's hypotheses are actually satisfied by (3.4) under (H1)-(H4): take p>2, f_i(s,µ)=λ|s|^{p-2}s, and a superlinear σ with the stated growth, and verify (2.3) with φ_4∈ℓ¹; a single violation would void the fixed-point step. For the rate, simulate the linear additive-noise system (5.49) on a finite chain, estimate W_1(L(z^ε(τ)), δ_0) for ε=10^{-k}; if the distance decays faster than C√ε, the optimality claim (5.52) is contradicted.

Watch

Extended reading notes

Core claim

On the paper's own terms, the core discovery is Theorem 4.2: the cocycle Φ associated with system (2.8) has a unique D-pullback measure attractor A = {A(τ)} in P_θ(D_ρ), which is compact, invariant (Φ(t,τ)A(τ)=A(τ+t)), and pullback-attracting for every bounded family D satisfying the backward integrability condition. Under the stronger dissipativity (5.11), Theorem 5.2 shows the attractor is a singleton {µ^ε_τ} forming an evolution family of pullback-mixing measures. Theorem 5.3 then states W_θ(µ^ε_τ, µ^0_τ) ≤ C_{τ,θ} ε^{1/2}, with exponent 1/2 optimal; optimality is shown by a linear additive-Lévy-noise equation whose explicit evolution family has W_1 distance to the deterministic Dirac law

Load-bearing premise

The entire construction rests on the imported well-posedness theorem for the distribution-frozen auxiliary equation (3.4), whose hypotheses are not restated or verified; in particular, the monotonicity condition (2.3) with φ_4 ∈ ℓ¹ must hold for the frozen equation to admit the unique solution used in the fixed-point step, and if it fails nothing in the cocycle construction has a foundation; a secondary chokepoint is the existential ε* in (4.39), since every attractor stateme

Editorial extensions

If this is right

If Theorem 4.2 is correct, every bounded family of initial laws with the required backward decay is pulled into the same compact invariant family A(τ); long-time behaviour of the delay system's law is therefore described by a single object.
If Theorem 5.3 is correct, the laws at noise level ε are within O(√ε) of the deterministic laws in W_θ, and no faster convergence can be guaranteed in general.
For distribution-independent versions of the system, the compactness lemma provides a direct route to existence of invariant measures for delay equations with superlinear Lévy noise, without higher-order moment estimates.
The methods are designed to cover a wide class of stochastic delay lattice equations, with either Brownian or Lévy noise or both, as long as the structural assumptions (H1)-(H4) hold.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Editorial inference: the √ε bound should hold for any mean-field delay lattice system whose noise enters additively or through Lipschitz-superlinear coefficients with the same scaling, since the fluctuation is a martingale of size √ε; the rate is likely intrinsic, not an artifact of the p-Laplace structure.
Editorial inference: the t≥ρ continuity obstruction suggests that numerical or data-driven schemes that couple the chain at time distances shorter than the delay may observe apparent non-Feller behaviour; a concrete extension would test whether the attractor convergence also holds in the uniform metric on continuous segments as the jump intensity tends to zero with bounded variation.
Editorial inference: for time-periodic forcing, the singleton pullback measure should become a periodic evolution family; checking whether the constant C_{τ,θ} in (5.45) is uniformly bounded in τ would indicate whether the convergence is exponentially stable in time, a question the paper leaves open.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague — here is the take: this is the first measure-attractor theory for McKean-Vlasov delay lattice systems with superlinear Lévy noise, and the √ε vanishing-noise rate with a genuine lower-bound example is the right headline. Structurally the paper is serious and honestly scoped. It discloses that the cocycle is only continuous for t ≥ ρ, routes the attractor invariance through that continuity, and restricts every attractor statement to ε < ε*. The Skorohod-space machinery for delay plus jumps is real work; the Jakubowski–Aldous compactness argument is a genuine technical departure from earlier Brownian treatments; and the rate theorem is derived from stated assumptions, not fitted to a target.

The main soft spot is exactly the one the stress-test note identifies. Theorem 3.1 imports well-posedness of the frozen equation (3.4) from [58, Theorem 3.1] together with [15] without restating those hypotheses or verifying that (H1)–(H4) imply them. The frozen coefficients depend on time through the càdlàg path µ(t) and may be only right-continuous; if [58] assumes more, the import does not close. That step is load-bearing: every later object — the cocycle, absorbing sets, tightness, attractor, rate — is defined through solutions of (2.8). I do not think this is fatal; it is a missing verification, likely patchable. But as written, the foundation is black-boxed.

Secondary issues are proportionate. ε* is existential from (4.39), so all results hold only for sufficiently small noise with no quantitative bound. The optimality example is linear additive Lévy noise, which sits outside the superlinear hypotheses; so 'optimal' is a lower bound for a special case, not a universal sharpness statement across the theorem's hypotheses. The density of estimates in Sections 3–5 means a referee will need to do line-level audit; I found no circularity — the constants are chosen by sign conditions, not by fitting a conclusion.

Who should read this: anyone working on measure attractors, McKean-Vlasov SPDEs, or lattice dynamics with jumps. It deserves a serious referee. My recommendation: send to peer review, but insist the authors either prove well-posedness for (3.4) or explicitly verify the cited hypotheses, and scope the optimality claim.

Referee Report

3 major / 4 minor

Summary. The paper studies the McKean–Vlasov stochastic p-Laplace lattice system with time delay and Lévy noise, Eq. (2.8). It claims existence and uniqueness of càdlàg solutions (Theorem 3.1), and then constructs a non-autonomous cocycle on probability measures over the Skorohod segment space D_ρ. Under dissipativity and integrability hypotheses (H1)–(H4), (3.3), (4.22), (4.26), it proves existence of a unique D-pullback measure attractor in P_θ(D_ρ) (Theorem 4.2). The proof handles the lack of all-time continuity by establishing continuity only for t ≥ ρ and uses tail estimates plus Jakubowski–Aldous tightness for asymptotic compactness. In the final section, the paper proves upper semi-continuity of the attractor as ε→0 (Theorem 5.1), shows that under stronger dissipativity the attractor is a singleton (Theorem 5.2), and derives the Wasserstein rate W_θ(µ^ε_τ, µ^0_τ) ≤ C ε^{1/2}, claimed optimal (Theorem 5.3).

Significance. If the proof gaps identified below are repaired, this is a substantial contribution. The paper extends measure-attractor theory to distribution-dependent delay lattice equations with jumps, a combination not previously treated. The use of the Skorohod topology for the segment process, the careful routing around the lack of continuity for t<ρ in the invariance proof, the higher-order moment and tail estimates, and the explicit rate of convergence are all valuable. The authors also provide explicit dissipativity conditions and an analytic argument for the √ε rate, which is a serious strength. The central architecture is standard pullback-attractor theory, but the technical difficulties specific to superlinear Lévy noise and delayed McKean–Vlasov structure are addressed in detail.

major comments (3)

[Section 3, proof of Theorem 3.1 (after Eq. (3.4))] The existence and uniqueness of the frozen equation (3.4) is imported without verification: 'Similar to the arguments in [58, Theorem 3.1] together with [15].' This is load-bearing because every later object — the cocycle Φ, the absorbing sets, the asymptotic compactness, the attractor, and the rate results — is defined via solutions of (2.8), whose construction in Step 2 rests entirely on the frozen equation. The coefficients f_µ(t)(s)=f(s,µ(t)) and σ_µ(t)(s)=σ(s,µ(t)) depend on time through a càdlàg path µ∈D_φ([τ−ρ,τ+T],P_2(ℓ^2)), so it is not automatic that [58, Theorem 3.1] applies. In particular, the paper neither states the hypotheses of [58] nor verifies that (H1)–(H4) imply them for coefficients that are only right-continuous in time. The one-sided Lipschitz structure after freezing also needs checking. Please either restate [58, Theorem 3.1] and verify all its hypotheses for (3.
[§5.2, Theorem 5.3, Step 2 (Eqs. (5.49)–(5.52))] The optimality example is not inside the theorem's assumptions. System (5.49) sets f=0, σ=0, and takes A to be the linear discrete Laplacian, so the p-Laplace structure and the superlinear dissipativity condition (2.1) with p>2, λ_1>0 are absent. Thus the lower bound C_1√ε for the linear additive-Lévy system does not exclude the possibility that every system satisfying (H1)–(H4), (3.2), (3.3), (4.22), (4.26), and (5.11) has a rate better than √ε. To claim that the order 1/2 is optimal within the stated class, the counterexample must satisfy all hypotheses of Theorem 5.3, or the optimality claim must be weakened. At minimum, an approximation argument or a nonlinear example with explicit invariant family is needed.
[§2 (H4) and §3, Theorem 3.1] The paper assumes q∈[2,p) in (H4), but all main results are stated for θ∈(2, 2(p−2)/(q−2)). This interval is undefined when q=2. Since q=2 is explicitly allowed, the theorem statements are not meaningful for an admissible parameter choice. The hypotheses should either be changed to q>2, or the formula for the upper bound should be clarified (e.g., interpreted as ∞ when q=2). This affects Theorem 3.1, Theorem 4.1, Theorem 4.2, and the results in Section 5.

minor comments (4)

[§5, before Theorem 5.2] The Wasserstein distance W_θ on P_θ(D_ρ) is used in Theorem 5.2 and Theorem 5.3 but is never defined. Earlier the paper defines W_m on P_m(ℓ^2) and the Fortet–Mourier-type metric d_P on P(D_ρ). Please define W_θ on the segment space D_ρ and state the coupling inequality used in (5.46).
[Section 3, Step 2] The space D_φ([τ−ρ,τ+T],P_2(ℓ^2)) with the weighted sup metric is called 'complete' without proof. Completeness of the space of càdlàg paths with sup metric is standard but should be stated, especially because the delay segment imposes the constraint µ(t)=Lφ(t−τ) on [τ−ρ,τ].
[§5.2, Eq. (5.21)] There is a typo in the Lévy integral: the integral is written over y∈Z rather than y∈Y in the display after (5.21). This should be corrected for readability.
[General notation] The paper uses both 'c`adl`ag' and 'càdlàg' and has occasional typos such as 'esitmates' in the Section 3 title. A careful proofreading pass is recommended.

Circularity Check

0 steps flagged · score 1.0 of 10

No constructional circularity; the only flagged issue is an unverified self-citation for well-posedness, which is a foundation gap rather than a circularity.

full rationale

The derivation chain is largely self-contained from the explicit hypotheses (H1)-(H4), (3.3), (4.22), (4.26) and (5.11). The attractor and rate results are obtained by analytic estimates: the constants β, δ and ε* are chosen through the sign conditions (4.24)-(4.25), (4.39) and (5.13), not fitted to a target conclusion. The contraction argument in Step 2 of Theorem 3.1, the absorbing-set estimates, the tightness argument in Lemma 4.5, and the convergence estimates in Lemmas 5.1-5.3 are all derived inequalities; no quantity is defined in terms of the desired attractor or convergence rate. The lower bound in Theorem 5.3 is an independent linear additive-Lévy computation (5.49)-(5.52), so the optimality claim is not circular. The one load-bearing gap is in the proof of Theorem 3.1: well-posedness of the frozen equation (3.4) is imported as 'Similar to the arguments in [58, Theorem 3.1] together with [15]' without restating or verifying the hypotheses of the cited results. Since those references share authors with the present paper, this is a self-citation that is load-bearing for the existence of solutions, and if its hypotheses do not cover (H1)-(H4), Theorems 4.2 and 5.3 would lack a foundation. However, this is a verification/support gap, not a circular reduction: no equation is equal to its own input by construction, and no fitted parameter is renamed as a prediction. I therefore find no significant circularity, with a score of 1 reflecting the minor unverified self-citation concern.

Assumptions & free parameters 4 free parameters · 6 assumptions · 0 invented entities

The core contribution is honest: every theorem is conditional on explicitly stated structural hypotheses. The price of the paper is the stack of regularity/dissipativity assumptions (H1)-(H4), (3.2)-(3.3), (4.22)-(4.26), (5.11)-(5.13) — none of which contain the target results. The only externally imported machinery is the well-posedness of the frozen equation (3.4), credited to [58] and [15]. The model constants λ, λ₁, λ₂, p, q and the sequences φ_i, L_F, L are inputs from (H1)-(H4), not free parameters. No invented entities are introduced.

free parameters (4)

β (exponential dissipation rate) = analytically chosen in (0,1) via (4.24)-(4.25)
Weight used in the pullback estimates; exists by (4.22). Not data-fitted.
δ (small constant in (5.13)) = arbitrarily small, in (0,1)
Introduced inside Lemma 5.3 to make Lévy-noise Itô remainders scale as ε^{θ/2} in (5.29)-(5.33).
ε* (max noise intensity) = non-explicit, ε*(θ,β) ∈ (0,1), from (4.37)-(4.39)
All main theorems are confined to ε ∈ (0,ε*). The threshold is existential only.
ε₀ (rate-regime threshold) = non-explicit, ε₀(θ) ≤ ε*
Window for the contraction in Lemma 5.2 and the rate estimate in Lemma 5.3.

assumptions (6)

domain assumption Structural hypotheses (H1)-(H4): monotone superlinear drift f (conditions (2.1)-(2.3)), Lipschitz delay coupling F, superlinear noise coefficients σ_k, σ̃ with growth index q ∈ [2,p), and summability of the coefficient sequences.
Section 2; defines the admissible equation class. Stated, not derived.
domain assumption Higher-moment integrability (3.2)-(3.3) of g, h, h̃ and of the Lévy kernels Σ L̃_{σ,i} and |σ̃_i(0,δ_0,·)| up to order θ.
Needed for the θ-th moment estimates in Theorem 3.1, Step 4.
ad hoc to paper Dissipativity condition (4.22): λ > ‖φ_1‖_{ℓ∞} + ‖φ_1‖_1 + 4θ^{-1}((θ-1)^{(θ-1)/θ})(‖L_F‖_{ℓ∞} + ‖L_F‖).
Tailored so that β, ε* exist making (4.24)-(4.25) and (4.39) hold; guarantees the pullback absorbing set.
ad hoc to paper Pullback integrability (4.26) of the forcing data with exponential weight e^{-β(τ-s)}.
Ensures the absorbing radius R_τ in Lemma 4.2 is finite.
ad hoc to paper Stronger dissipativity (5.11)-(5.13) used only for Lemma 5.2, Theorems 5.2 and 5.3 (singleton attractor and rates).
Yields the exponential contraction and the ε^{θ/2} error structure via the small constant δ.
standard math Standard stochastic-analysis tools: Itô formula, BDG inequality, Gronwall inequality, Fatou's lemma, Skorokhod representation, Jakubowski's tightness theorem, Aldous's criterion.
Invoked throughout Sections 3-5 without proof; standard background.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Existence and vanishing noise limit of measure attractors for McKean-Vlasov $p$-Laplacian lattice systems with delay driven by L\'evy noise." pith.science (2026). https://pith.science/paper/MXTVWUKD

@misc{pith2026260711773,
  author       = {Pith},
  title        = {Pith review of: Existence and vanishing noise limit of measure attractors for McKean-Vlasov $p$-Laplacian lattice systems with delay driven by L\'evy noise},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MXTVWUKD}},
  note         = {Machine review of arXiv:2607.11773}
}

abstract

This paper is concerned with the existence and the limiting behavior of measure attractors of distribution laws of the solution segment process for the McKean-Vlasov stochastic $p$-Laplace lattice system with time delay driven by L\'evy noise. The nonlinear drift and diffusion terms are allowed to have superlinear growth. Due to time delay, the Skorohod metric space is employed to describe the trajectories of the solutions with jumps. We first prove the existence and uniqueness of c\`adl\`ag solutions for the lattice system, and then define a non-autonomous cocycle acting on the Borel probability measures in the Skorohod space. This cocycle is continuous in bounded subsets of the space of probability measures only when time is sufficiently large. We then prove the existence of pullback absorbing sets and the asymptotic compactness of the cocycle as well as the existence and uniqueness of pullback measure attractors. We finally investigate the limiting behavior of measure attractors of the lattice system as the noise intensity approaches zero, and establish the optimal convergence rate of singleton measure attractors in the Wasserstein distance of order $\theta$.

Review history (2 revisions) →

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

63 extracted references · 2 linked inside Pith

[15]

R. Wang, B. Wang, Random dynamics ofp-Laplacian lattice systems driven by infinite- dimensional nonlinear noise, Stoch. Process. Appl.130(2020) 7431–7462

2020
[58]

Z. Chen, D. Yang, S. Zhong, Weak mean attractor and periodic measure for stochastic lattice systems driven by L´ evy noises, Stoch. Anal. Appl.41(2023) 509–544

2023
[1]

J. I. Diaz, F. D. Th´ elin, On a nonlinear parabolic problem arising in some models related to turbulent flows, SIAM J. Math. Anal.25(4)(1994) 1085–1111

1994
[2]

J. R. Esteban, J. L. V´ azquez, Homogeneous diffusion inRwith power-like nonlinear diffusivity, Arch. Rational Mech. Anal.103(1)(1988) 39–80

1988
[3]

Schmitz, A

K. Schmitz, A. Zimmermann, The stochasticp-Laplace equation onR d. Stoch. Anal. Appl.41(5) (2023) 892–917

2023
[4]

Wang, Well-posedness of fractional stochasticp-Laplace equations driven by superlinear trans- port noise, Discrete Contin

B. Wang, Well-posedness of fractional stochasticp-Laplace equations driven by superlinear trans- port noise, Discrete Contin. Dyn. Syst.-B34(2026) 471–507

2026
[5]

Liu, On the stochasticp-Laplace equation, J

W. Liu, On the stochasticp-Laplace equation, J. Math. Anal. Appl.360(2)(2009) 737–751. 72

2009
[6]

B. Gess, J. T¨ olle, Ergodicity and local limits for stochastic local and nonlocalp-Laplace equations, SIAM J. Math. Anal.48(6)(2016) 4094–4125

2016

Show all 63 references

[7]

R. Wang, T. Caraballo, N. H. Tuan, Mean attractors and invariant measures of locally monotone and generally coercive SPDEs driven by superlinear noise, J. Differ. Equ.381(2024) 209–259

2024
[8]

Brze´ zniak, W

Z. Brze´ zniak, W. Liu, J. Zhu, Strong solutions for SPDE with locally monotone coefficients driven by L´ evy noise. Nonlinear Anal.: Real World Appl.17(2014) 283–310

2014
[9]

Neelima, On well-posedness of stochastic anisotropicp-Laplace equation driven by L´ evy noise, Potential Anal.57(2022) 671–707

2022
[10]

Kavin, A.K

R. Kavin, A.K. Majee, Nonlinear SPDE driven by L´ evy noise: well-posedness, optimal control and invariant measure, J. Evol. Equ.25(2025) 22

2025
[11]

L.O. Chua, T. Roska, The CNN paradigm, IEEE Trans. Circuits Syst.40(1993) 147–156

1993
[12]

S.N. Chow, J. Mallet-Paret, Pattern formation and spatial chaos in lattice dynamical systems I, IEEE Trans. Circuits Syst.42(1995) 746–751

1995
[13]

Majda, X.T

A.J. Majda, X.T. Tong, Geometric ergodicity for piecewise contracting processes with applications for tropical stochastic lattice models, Commun. Pure Appl. Math.69(2016) 1110–1153

2016
[14]

Christodoulides, F

D. Christodoulides, F. Lederer, Y. Silberberg, Discretizing light behaviour in linear and nonlinear waveguide lattices, Nature424(2003) 817–823

2003
[16]

F.L. Wang, B. Guo, Enlarged evolution system of measures of stochasticp-Laplace lattice systems with superlinear noise, Nonlinearity38(4)(2025) 045005

2025
[17]

W. Zhao, X. Liu, Invariant sample measures for non-autonomous stochastic non-local discrete p-Laplacian equations with time varying delay and multiplicative noises, Appl. Math. Optim.91 (2025) 1–40

2025
[18]

Caraballo, F

T. Caraballo, F. Morillas, J. Valero, On differential equations with delay in Banach spaces and attractors for retarded lattice dynamical systems, Discrete Contin. Dyn. Syst.34(2014) 51–77

2014
[19]

Y. Li, S. Yang, T. Caraballo, Optimization and convergence of numerical attractors for discrete- time quasi-linear lattice system, SIAM J. Numer. Anal.61(2023) 905–928

2023
[20]

J. Xu, T. Caraballo, J. Valero, Dynamics and Wong-Zakai approximations of stochastic nonlocal PDEs with long time memory, Qual. Theory Dyn. Syst.23(2024) 228

2024
[21]

Z. Chen, B. Wang, Long-term dynamics of fractional stochastic delay reaction-diffusion equations on unbounded domains, Stoch. Partial Differ. Equ.: Anal. Comput.13(2025) 180–242

2025
[22]

Buckdahn, J

R. Buckdahn, J. Li, S. Peng, C. Rainer, Mean-field stochastic differential equations and associated PDEs, Ann. Probab.45(2)(2017) 824–878. 73

2017
[23]

Cardaliaguet, F

P. Cardaliaguet, F. Delarue, J. M. Lasry, P. L. Lions,The Master Equation and the Convergence Problem in Mean Field Games, Princeton University Press, 2019

2019
[24]

W. Liu, L. Wu, C. Zhang, Long-time behaviors of mean-field interacting particle systems related to Mckean-Vlasov equations, Commun. Math. Phys.387(1)(2021) 179–214

2021
[25]

Ahmed, X

N.U. Ahmed, X. Ding, A semilinear Mckean-Vlasov stochastic in Hilbert space, Stoch. Process. Appl.60(1)(1995) 65–85

1995
[26]

Wang, Distribution dependent SDEs for Laudau type equations, Stoch

F.Y. Wang, Distribution dependent SDEs for Laudau type equations, Stoch. Process. Appl.128 (2018) 595–621

2018
[27]

Wang, P.P

F.Y. Wang, P.P. Ren,Distribution Dependent Stochastic Differential Equations, World Scientific Publishing, 2025

2025
[28]

Hammersley, S

W. Hammersley, S. David, L. Szpruch, Weak existence and uniqueness for McKean-Vlasov SDEs with common noise, Ann. Probab.49(2021) 527–555

2021
[29]

R¨ ockner, X

M. R¨ ockner, X. Zhang, Well-posedness of distribution dependent SDEs with singular drifts, Bernoulli27(2021) 1131–1158

2021
[30]

Liang, M.B

M. Liang, M.B. Majka, J. Wang, Exponential ergodicity for SDEs and McKean-Vlasov processes with L´ evy noise, AIHP-Probab. Stat.57(2021) 1665–1701

2021
[31]

Wang, Exponential ergodicity for non-dissipative McKean-Vlasov SDEs, Bernoulli29(2023) 1035–1062

F.Y. Wang, Exponential ergodicity for non-dissipative McKean-Vlasov SDEs, Bernoulli29(2023) 1035–1062

2023
[32]

W. Hong, S. Li, W. Liu, Large deviation principle for McKean-Vlasov quasilinear stochastic evolution equations, Appl. Math. Optim.84(2021) 1119–1147

2021
[33]

W. Liu, Y. Song, J. Zhai, T. Zhang, Large and moderate deviation principles for McKean-Vlasov SDEs with jumps, Potential Anal.59(2023) 1141–1190

2023
[34]

Huang, M

X. Huang, M. R¨ ockner, F.Y. Wang, Nonlinear Fokker-Planck equations for probability measures on path space and path-distribution dependent SDEs, Discrete Contin. Dyn. Syst.39(2019) 3017–3035

2019
[35]

J. Bao, M. Scheutzow, C. Yuan, Existence of invariant probability measures for functional McKean-Vlasov SDEs, Electron. J. Probab.27(2022) 1–14

2022
[36]

L. Shi, J. Shen, K. Lu, Pullback measure attractors and limiting behaviors of McKean-Vlasov stochastic delay lattice systems, arXiv:2412.15528

arXiv
[37]

Schmalfuß, Long-time behaviour of the stochastic Navier-Stokes equation, Math

B. Schmalfuß, Long-time behaviour of the stochastic Navier-Stokes equation, Math. Nachr.152 (1991), 7–20

1991
[38]

Capi´ nski, N.J

M. Capi´ nski, N.J. Cutland, Measure attractors for stochastic Navier-Stokes equations, Electron. J. Probab.3(8)(1998) 1–15

1998
[39]

D. Li, B. Wang, Pullback measure attractors for non-autonomous stochastic reaction-diffusion equations on thin domains, J. Differ. Equ.397(2024) 232–261. 74

2024
[40]

Z. Chen, B. Wang, S. Zhong, Existence, upper semicontinuity and convergence rate of measure attractors for non-autonomous superlinear stochastic Schr¨ odinger delay lattice systems, J. Non- linear Sci.35(2025) 119

2025
[41]

L. Shi, J. Shen, B. Wang, Long-time dynamics of McKean-Vlasov stochastic reaction-diffusion equations onR n, J. Differ. Equ.453(2026) 113855

2026
[42]

F. Bai, Z. Chen, X. Sun, Existence and approximation of measure attractors and invariant mea- sures for McKean-Vlasov stochastic lattice system with L´ evy noise, J. Differ. Equ.453(2026) 113784

2026
[43]

Y. Wang, F. Wu, X. Mao, Stability in distribution of stochastic functional differential equations, Syst. Control Lett.132(2019) 104513

2019
[44]

Y. Lin, D. Li, Limiting behavior of invariant measures of highly nonlinear stochastic retarded lattice systems, Discrete Contin. Dyn. Syst.-B,27(12)(2022) 7561–7590

2022
[45]

Z. Chen, X. Sun, B. Wang, Invariant measures and large deviation principles for stochastic Schr¨ odinger delay lattice systems, Proc. Royal Soc. Edinburgh A: Math.155(5)(2025) 1813– 1854

2025
[46]

Jakubowski, On the Skorokhod topology, AIHP-Probab

A. Jakubowski, On the Skorokhod topology, AIHP-Probab. Stat.22(3)(1986) 263–285

1986
[47]

Aldous, Stopping times and tightness, Ann

D. Aldous, Stopping times and tightness, Ann. Probab.6(2)(1978) 335–340

1978
[48]

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

2006
[49]

L. Bo, C. Yuan, Stochastic delay differential equations with jump reflection: invariant measure, Stochastics88(6)(2016) 841–863

2016
[50]

S. Mi, R. Li, D. Li, Pullback measure attractors for non-autonomous fractional stochastic reaction- diffusion equations on unbounded domains, Appl. Math. Optim.90(2024) 55

2024
[51]

Z. Chen, B. Wang, Wasserstein convergence of invariant measures for fractional stochastic reaction-diffusion equations on unbounded domains, Appl. Math. Lett.147(2024) 108842

2024
[52]

Da Prato, M

G. Da Prato, M. R¨ ockner, A note on evolution systems of measures for time-dependent stochastic differential equations, In:Seminar on Stochastic Analysis, Random Fields and Applications V, Prog. Probab., Vol. 59, Birkh¨ auser, 2007, 115–122

2007
[53]

R. Liu, K. Lu, Exponential mixing and limit theorems of quasi-periodically forced 2D stochastic Navier-Stokes equations in the hypoelliptic setting, Commun. Math. Phys.406(2025) 55

2025
[54]

R. Wang, T. Caraballo, N. Tuan, Asymptotic stability of evolution systems of probability mea- sures for nonautonomous stochastic systems: theoretical results and applications, Proc. Am. Math. Soc.151(2023) 2449–2458

2023
[55]

Applebaum,L´ evy Processes and Stochastic Calculus, Cambridge Univere Press, 2009

D. Applebaum,L´ evy Processes and Stochastic Calculus, Cambridge Univere Press, 2009

2009
[56]

Billingsley,Convergence of Probability Measures, Wiley & Sons, 2013

P. Billingsley,Convergence of Probability Measures, Wiley & Sons, 2013. 75

2013
[57]

van der Vaart and J

A. van der Vaart and J. Wellner,Weak Convergence and Empirical Processes with Applications to Statistics, Springer, 2023

2023
[59]

Peszat, J

S. Peszat, J. Zabczyk,Stochastic Partial Differential Equations with L´ evy Noise: An Evolution Equation Approach, Cambridge University Press, 2007

2007
[60]

Barczy, Z

M. Barczy, Z. Li, G. Pap, Yamada–Watanabe results for stochastic differential equations with jumps, Internat. J. Stoch. Anal.2015(2015) 460472

2015
[61]

Fahim, E

K. Fahim, E. Hausenblas, K.H. Karlsen, Yamada-Watanabe uniqueness results for SPDEs driven by Wiener and pure jump processes, arXiv:2501.02924

arXiv
[62]

Wang, Sufficient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems, J

B. Wang, Sufficient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems, J. Differ. Equ.253(2012) 1544–1583

2012
[63]

Wooster, Evolution systems of measures for non-autonomous Ornstein-Uhlenbeck processes with L´ evy noise, Commun

R. Wooster, Evolution systems of measures for non-autonomous Ornstein-Uhlenbeck processes with L´ evy noise, Commun. Stoch. Anal.5(2011) 353–370. 76

2011

Pith tools

Reviewed August 2, 2026 · model on record in the stance chip above.

[1] [15]

R. Wang, B. Wang, Random dynamics ofp-Laplacian lattice systems driven by infinite- dimensional nonlinear noise, Stoch. Process. Appl.130(2020) 7431–7462

2020

[2] [58]

Z. Chen, D. Yang, S. Zhong, Weak mean attractor and periodic measure for stochastic lattice systems driven by L´ evy noises, Stoch. Anal. Appl.41(2023) 509–544

2023

[3] [1]

J. I. Diaz, F. D. Th´ elin, On a nonlinear parabolic problem arising in some models related to turbulent flows, SIAM J. Math. Anal.25(4)(1994) 1085–1111

1994

[4] [2]

J. R. Esteban, J. L. V´ azquez, Homogeneous diffusion inRwith power-like nonlinear diffusivity, Arch. Rational Mech. Anal.103(1)(1988) 39–80

1988

[5] [3]

Schmitz, A

K. Schmitz, A. Zimmermann, The stochasticp-Laplace equation onR d. Stoch. Anal. Appl.41(5) (2023) 892–917

2023

[6] [4]

Wang, Well-posedness of fractional stochasticp-Laplace equations driven by superlinear trans- port noise, Discrete Contin

B. Wang, Well-posedness of fractional stochasticp-Laplace equations driven by superlinear trans- port noise, Discrete Contin. Dyn. Syst.-B34(2026) 471–507

2026

[7] [5]

Liu, On the stochasticp-Laplace equation, J

W. Liu, On the stochasticp-Laplace equation, J. Math. Anal. Appl.360(2)(2009) 737–751. 72

2009

[8] [6]

B. Gess, J. T¨ olle, Ergodicity and local limits for stochastic local and nonlocalp-Laplace equations, SIAM J. Math. Anal.48(6)(2016) 4094–4125

2016