Existence of Periodic and Stationary Solutions to Distribution-Dependent SDEs

Ethan Wong; Wei Sun

REVIEW 2 major objections 4 minor 1 cited by

Existence of Periodic and Stationary Solutions to Distribution-Dependent SDEs

T0 review · 2 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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

Pith's one-line read Distribution-dependent SDEs with periodic coefficients admit periodic-in-law solutions whenever a Lyapunov function controls the drift, and stationary solutions in the time-homogeneous case.

desk verdict Solid new existence criterion for theta-periodic McKean-Vlasov solutions, but the advertised stationary corollary is stated without its proof. read the letter →

arxiv 2501.09176 v1 pith:A2G7B6ZW submitted 2025-01-15 math.PR

Wei Sun , Ethan Wong This is my paper · ORCID

classification math.PR MSC 60H1037A50

keywords distribution-dependentSDEMcKean-VlasovperiodicsolutionstationaryLyapunovfunctionweakconvergenceWassersteinmetric

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 proves that a distribution-dependent stochastic differential equation with periodic coefficients has a periodic-in-law solution whenever the coefficients admit a Lyapunov function with a drift bound, provided the one-step semigroup is well behaved on a Wasserstein space. This matters because most existing long-time results for McKean–Vlasov equations are time-homogeneous, and the nonlinearity of the distribution-dependent semigroup blocks the standard fixed-point routes to periodic solutions. The authors give three distinct Lyapunov conditions — a polynomial drift bound, a stronger interacting bound, and a pure supermartingale condition — and verify them on concrete models including a homogeneous Landau equation.

What carries the argument

The central object is the time-θ semigroup $P_\theta^*$ acting on the Polish space $(P_\vartheta(\mathbb{R}^d), W_\vartheta)$ of probability measures with finite ϑ-th moment, equipped with the Wasserstein-ϑ metric. For distribution-dependent equations this semigroup is generally nonlinear, so the proof cannot appeal to linear Markov semigroup ergodic theory; instead the Lyapunov function V and its generator $\mathcal{L}V$ supply the drift inequalities that make the Cesàro averages of $\{P_s^*\delta_0\}$ tight, and condition (H2) — continuity of $P_\theta^*$ in $W_\vartheta$ — is exactly what turns the weak convergence of those averages into the fixed-point equation $P_\theta^*\nu = \nu$. For condition (H3c), the same semigroup is shown to map a compact convex set $K = \{\mu : \int V(0,x)\,\mu(dx) \le |V(0,0)|\}$ into itself continuously, so Schauder's fixed point theorem applies.

What would settle it

Find coefficients $b,\sigma$ satisfying (H0) and (H1) and one of (H3a)–(H3c), but for which $P_\theta^*$ is discontinuous at some $\mu \in P_\vartheta(\mathbb{R}^d)$ in the $W_\vartheta$ metric; then Theorem 1.3 would predict a periodic solution but the proof's fixed-point step breaks down. Concretely, compute two initial laws $\mu_n \to \mu$ in $W_\vartheta$ and check whether $W_\vartheta(P_\theta^*\mu_n, P_\theta^*\mu)$ fails to go to zero.

Watch

Extended reading notes

Core claim

The central claim is Theorem 1.3: under periodicity of coefficients (H0), strong existence and uniqueness (H1), and Wasserstein-θ continuity of the time-θ semigroup (H2), each of the Lyapunov conditions (H3a), (H3b), or (H3c) implies the existence of a θ-periodic measure ν in $P_\vartheta(\mathbb{R}^d)$ with $P_\theta^*\nu = \nu$, and hence a θ-periodic solution whose finite-dimensional distributions are invariant under time shifts by θ. The proof splits: for (H3a) and (H3b) it runs a Krylov–Bogoliubov argument on the Cesàro averages of the law of the solution started at 0, extracting a weak limit and using (H2) to show the limit is periodic; for (H3c) it constructs a compact convex set of laws on which $P_\theta^*$ acts continuously and applies the Schauder fixed point theorem. Corollary 1.5 gives the time-homogeneous analogue: a stationary solution exists if the coefficients do not depend on t and the same Lyapunov conditions hold with t omitted.

Load-bearing premise

The load-bearing premise is condition (H2): the one-step semigroup $P_\theta^*$ must map $P_\vartheta$ into itself continuously with respect to the Wasserstein-ϑ metric; for distribution-dependent equations this continuity is not automatic, and if it fails the Krylov–Bogoliubov or Schauder argument does not close.

Editorial extensions

If this is right

If Theorem 1.3 is correct, every DDSDE with θ-periodic coefficients satisfying (H0)–(H2) and one of the three Lyapunov inequalities has at least one θ-periodic solution; uniqueness is not claimed.
Corollary 1.5 gives existence of a stationary solution for time-homogeneous DDSDEs under the same type of conditions, without requiring the coefficient maps to be contractions.
The results cover coefficients that are not necessarily monotone in the distribution variable, as long as the Lyapunov inequality holds.
The examples include a concrete nonlinear SDE (Example 3.2) and a time-dependent homogeneous Landau equation (Example 3.5), showing the criteria apply beyond additive noise.

Reading between the lines

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

The proof's reliance on (H2) suggests that the periodic measure may fail to exist if the semigroup is only continuous in a weaker topology; a natural test is to look for DDSDEs where Cesàro averages converge weakly but $P_\theta^*$ is discontinuous at the limit.
Because uniqueness is not established, the framework could be extended to phase-transition phenomena where multiple periodic measures coexist, analogous to known non-uniqueness results for stationary measures of McKean–Vlasov SDEs.
The Lyapunov conditions are stated for the solution from a deterministic start; one could try weakening them to require only a bound on the Cesàro averages of the law.

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

The periodic half of this paper is a real, mostly correct result; the stationary half is advertised but not proved. Sun and Wong prove existence of theta-periodic solutions for time-inhomogeneous McKean-Vlasov SDEs under Lyapunov conditions, using Krylov-Bogoliubov weak convergence plus a Schauder fixed point argument for the supermartingale-type condition H3c. The proof is careful and, as far as I checked, the steps hold: the moment bounds from Ito's formula are clean, the Cesaro averages have uniform r-th moments and hence are tight, convergence in W_theta follows from Villani's criteria, and H2 does the job of passing P^*_theta through the limit. The nonlinearity of the semigroup is handled honestly, and the Schauder setup with the Kantorovich-Rubinstein norm is legitimate. The examples are concrete and show the criteria are checkable; the Landau equation example is a nice test.

The soft spot is exactly what the stress-test says. Corollary 1.5 is a stated main result, but Section 2 ends with 'The proof of Corollary 1.5 is very similar, which we will omit.' It is not a one-line special case of Theorem 1.3. The periodic theorem only gives a measure invariant under P^*_theta for one fixed period; stationarity needs P^*_t invariance for every t > 0. That requires the usual shift/diagonalization argument to get one measure that works for all t simultaneously. That argument may be standard, but it is not included, and Remark 2.4 asserts the conclusion without proof. A referee should require a written proof before the paper is accepted. I don't see a false step in the periodic proof, so this is a completeness problem, not a correctness problem.

The assumptions H2/H2' are imported from Wang's theorem; they are not automatic for nonlinear semigroups, but the paper cites the right result and checks it in the examples. The distinction from Bao-Reis-Wu's random periodic solutions is clearly drawn, and the citation pattern looks fair.

Who is this for: people working on long-time behavior of McKean-Vlasov equations. It deserves a serious referee. Send it out, but the referee report should insist on the proof of Corollary 1.5, or a revision that downgrades it to a conjecture or remark.

Referee Report

2 major / 4 minor

Summary. The manuscript establishes criteria for the existence of θ-periodic and stationary solutions for distribution-dependent SDEs (McKean-Vlasov equations). Under structural assumptions (H0)-(H2), Theorem 1.3 asserts that any of three Lyapunov-type conditions (H3a), (H3b), or (H3c) yields a θ-periodic solution, and Corollary 1.5 states the time-homogeneous analogue. Section 2 proves the H3a/H3b cases by a Krylov-Bogoliubov argument with Cesàro averages and the H3c case by a Schauder fixed-point argument on a compact convex set of measures. Section 3 gives applications, including a time-dependent Landau-type example. The Schauder part and the reduction of periodic solutions to periodic measures in Lemma 2.2 are coherent, but the Cesàro part is invalid because it uses linearity of the nonlinear semigroup.

Significance. If the results were correct, the paper would provide flexible Lyapunov criteria for periodic solutions of time-inhomogeneous McKean-Vlasov SDEs, a topic with few existing tools, and the Landau-type example would be a useful application. The manuscript is also careful to allow non-unique stationary and periodic measures. However, the H3a/H3b half of Theorem 1.3 and the corresponding half of Corollary 1.5 rest on an unjustified linearity step, so the main advertised result is not established. The H3c/Schauder argument and the examples built on it, such as Example 3.7, are not affected by this defect.

major comments (2)

[Section 2, after Eq. (2.6)] The displayed identity (P^*_θ ν_n)(G) = (1/T_n)∫_0^{T_n}(P^*_{0,θ}P^*_{0,s}δ_0)(G)ds treats P^*_{0,θ} as if it commuted with convex combinations of initial laws. This is exactly the linearity that the authors show fails for DDSDEs in Section 1: in the example dX_t = Var(X_t)dt + dB_t, they compute P^*_t(ν0) ≠ ∫P^*_t δ_x ν0(dx). Starting the SDE from the mixture ν_n does not produce the mixture of the laws obtained from the components P^*_{0,s}δ_0, because the coefficients at time u depend on the global law of the solution at time u. Consequently the weak-convergence step (2.7) and the conclusion P^*_θ ν = ν are not justified for the H3a/H3b cases; the Krylov-Bogoliubov argument does not close. The H3c/Schauder proof is independent and appears sound.
[Section 2, end of proof of Theorem 1.3; Corollary 1.5] Corollary 1.5 is advertised as a main result, but its proof is omitted with the sentence 'The proof of Corollary 1.5 is very similar, which we will omit.' The conclusion requires a measure ν with P^*_t ν = ν for every t > 0, whereas Theorem 1.3 only produces P^*_θ ν = ν for one fixed θ. The natural adaptation of the Cesàro argument must be written out and, in particular, one must show that a single limit measure works for all h simultaneously; the manuscript supplies no such argument. Remark 2.4 also asserts the existence of an invariant measure under the assumptions of Corollary 1.5 without proof. Even setting aside the linearity issue above, this is a load-bearing gap in a central advertised claim.

minor comments (4)

[Example 3.2, (A4) case] The line 'Thus, condition (H3a) holds' is too quick: the displayed upper bound contains a positive K_{σ,1}|x|^2 term, and one must invoke Young's inequality with r > 2 to absorb it into -K_{b,4}|x|^r before the bound has the form required by (H3a).
[Example 3.5] In the definition of σ(t,x,μ), the integral is written over R^d, although the example is set in R^3; the domain should be R^3.
[Section 2, Cesàro computation] The notation P^*_{s+θ}δ_0 is used in the displayed computation for P^*_θ ν_n where elsewhere the notation means P^*_{0,s+θ}δ_0; please define this explicitly to avoid ambiguity.
[Abstract] There is a line-break typo in the first sentence ('dis tribution'); this should be corrected.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the derivation proceeds from explicit hypotheses via external compactness, weak-convergence, and Schauder fixed-point theorems; the omitted proof of Corollary 1.5 is a completeness gap, not a circular step.

full rationale

I find no circular step in the paper's derivation chain. Theorem 1.3 is proved from explicit hypotheses (H0)-(H2) plus one of (H3a)-(H3c), using standard external tools: Itô's formula, Cesàro averaging, weak-convergence compactness criteria ([35, Definition 6.8 and Theorem 6.9]; [11, Theorem 3.3.1]), and the Schauder fixed-point theorem ([31, Theorem 11.1.2]). The target measure ν is constructed as a weak and W_ϑ limit of Cesàro averages, not assumed or defined in terms of the conclusion. The constants in Lyapunov conditions are existential assumptions, not fitted parameters, and no quantity is 'predicted' from data fitted by the authors. Assumption H2 is genuinely assumed; in examples it is verified by appealing to Wang's external theorem [36, Theorem 2.1], not to the paper's own results. The paper's self-citations ([13], [14]) appear only as background references for periodic SDEs and are not load-bearing. The only flagged issue is an explicitly omitted proof: 'The proof of Corollary 1.5 is very similar, which we will omit' (end of Section 1 and Section 2). This is a real completeness gap — Corollary 1.5 requires P_t^*ν = ν for all t > 0, not just P_θ^*ν = ν — but it is a rigor/omission concern, not circularity, because the stationary conclusion is neither assumed nor fitted, and no equation reduces to an input by construction. Therefore the circularity score is 0.

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

The theorem rests on the existence and uniqueness and semigroup-continuity assumptions H1-H2, on one of three Lyapunov conditions, and on standard external theorems used to verify them. No numbers are fitted to data and no new entities are postulated; the constants in the Lyapunov inequalities are existential.

assumptions (4)

domain assumption H1: strong existence and strong/weak uniqueness in P_theta(R^d) for the DDSDE.
Stated on page 4; it guarantees the flow property (1.2), the semigroup P^*_{s,t}, and the equivalence in Lemma 2.2.
domain assumption H2: P^*_theta(P_theta(R^d)) is contained in P_theta(R^d) and P^*_theta is W_theta-continuous.
Stated on page 4 and used after (2.6) to pass W_theta-convergence through the period map; this is the weakest structural premise of the main proof.
domain assumption One of the Lyapunov conditions H3a, H3b, or H3c holds: existence of V in C^{1,2} with the stated lower bound and generator inequality.
These are the criteria of Theorem 1.3; the theorem is conditional on finding such a V.
standard math Wang's theorem [36, Theorem 2.1] and standard compactness and fixed point theorems.
Wang's theorem verifies H1-H2 in Remark 3.1 and the examples; Villani, Ethier-Kurtz, and Schauder results drive the proof in Section 2.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Existence of Periodic and Stationary Solutions to Distribution-Dependent SDEs." pith.science (2026). https://pith.science/paper/A2G7B6ZW

@misc{pith2026250109176,
  author       = {Pith},
  title        = {Pith review of: Existence of Periodic and Stationary Solutions to Distribution-Dependent SDEs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/A2G7B6ZW}},
  note         = {Machine review of arXiv:2501.09176}
}

read the original abstract

We investigate the periodic and stationary solutions of distribution-dependent stochastic differential equations. While generally, the semigroups associated with the equations are nonlinear, we show that the methods of weak convergence and Lyapunov functions can be combined to give efficient criteria for the existence of periodic and stationary solutions. Concrete examples are presented to illustrate the novel criteria.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Periodic solutions for McKean-Vlasov SDEs under periodic distribution-dependent Lyapunov conditions
math.DS 2025-01 reject novelty 5.0 of 10

Under periodic distribution-dependent Lyapunov conditions, the paper constructs T-periodic solutions for McKean-Vlasov SDEs by lifting the dynamics to the product space R^d times P(R^d).

Reference graph

Works this paper leans on

39 extracted references · 37 canonical work pages · cited by 1 Pith paper

[1]

N. U. Ahmed and X. Ding, On invariant measures of nonlinear Markov processes , Journal of Applied Mathematics and Stochastic Analysis 6 (1993), 385–406

work page 1993
[2]

J. Bao, G. D. Reis, and Y. Wu, The random periodic solutions for McKean-Vlasov stochas- tic diﬀerential equations , (2024), arXiv:2408.17242

arXiv 2024
[3]

J. Bao, M. Scheutzow, and C. Yuan, Existence of invariant probability measures for func- tional McKean-Vlasov SDEs , Electronic Journal of Probability 27 (2022), 1–14

work page 2022
[4]

V. I. Bogachev, M. R¨ ockner, and S. V. Shaposhnikov, Convergence in variation of solu- tions of nonlinear Fokker-Planck-Kolmogorov equations to stationary measures, Journal of Functional Analysis 276 (2019), 3681–3713

work page 2019
[5]

O. A. Butkovsky, On ergodic properties of nonlinear Markov chains and stocha stic McKean-Vlasov equations , Theory of Probability and Its Applications 58 (2014), 661– 674

work page 2014
[6]

Carmona and F

R. Carmona and F. Delarue, Probabilistic Theory of Mean Field Games with Applications , I and II , Springer, 2018

work page 2018
[7]

F. Chen, Y. Han, Y. Li, and X. Yang, Periodic solutions of Fokker-Planck equations , Journal of Diﬀerential Equations 263 (2017), 285–298

work page 2017
[8]

D. A. Dawson, Critical dynamics and ﬂuctuations for a mean ﬁeld model of co operative behaviour, Journal of Statistical Physics 41 (1983), 29–85

work page 1983

Show all 39 references

[9]

K. Du, Y. Jiang, and J. Li, Empirical approximation to invariant measures for McKean- Vlasov processes: mean-ﬁeld interaction vs self-interact ion, Bernoulli 29 (2023), 2492– 2518

2023
[10]

M. H. Duong and J. Tugaut, Stationary solutions of the Vlasov-Fokker-Planck equatio n: existence, characterization and phase-transition , Applied Mathematics Letters 52 (2016), 38–45

2016
[11]

S. N. Ethier and T. G. Kurtz, Markov Processes: Characterization and Convergence , Wiley, 2009

2009
[12]

G¨ artner, On the McKean-Vlasov limit for interacting diﬀusions , Mathematische Nachrichten 137 (1988), 197–248

J. G¨ artner, On the McKean-Vlasov limit for interacting diﬀusions , Mathematische Nachrichten 137 (1988), 197–248

1988
[13]

Guo and W

X. Guo and W. Sun, Periodic solutions of stochastic diﬀerential equations dr iven by L´ evy noises, Journal of Nonlinear Science 31 (2019)

2019
[14]

Guo and W

X. Guo and W. Sun, Periodic solutions of hybrid jump diﬀusion processes , Frontiers of Mathematics in China 16 (2021), 705–725. 19

2021
[15]

Hu and L

H. Hu and L. Xu, Existence and uniqueness theorems for periodic Markov proc ess and ap- plications to stochastic functional diﬀerential equation s, Journal of Mathematical Analysis and Applications 466 (2018), 896–926

2018
[16]

Huang, R

M. Huang, R. P. Malham´ e, and P. E. Caines, Large population stochastic dynamic games: closed-loop McKean-Vlasov systems and the Nash certainty e quivalence principle, Commu- nications in Information and Systems 6 (2006), 221–251

2006
[17]

Huang, R

M. Huang, R. P. Malham´ e, and P. E. Caines, Large-population cost-coupled LQG prob- lems with nonuniform agents: individual-mass behavior and decentralizedε-Nash equilibria, IEEE Transactions on Automatic Control 52 (2007), 1560–1571

2007
[18]

Kac, Foundations of kinetic theory , Proceedings of the 3rd Berkeley Symposium on Mathematical Statistics and Probability 3 (1956), 171–197

M. Kac, Foundations of kinetic theory , Proceedings of the 3rd Berkeley Symposium on Mathematical Statistics and Probability 3 (1956), 171–197

1956
[19]

Khasminskii, Stochastic Stability of Diﬀerential Equations , Springer, 2011

R. Khasminskii, Stochastic Stability of Diﬀerential Equations , Springer, 2011

2011
[20]

J. M. Lasry and P. L. Lions, Mean ﬁeld games. I. the stationary case , Comptes Rendus Hebdomadaires des S´ eances de l’Acad´ emie des Sciences343 (2006), 619–625

2006
[21]

J. M. Lasry and P. L. Lions, Mean ﬁeld games. II. ﬁnite horizon and optimal control , Comptes Rendus Hebdomadaires des S´ eances de l’Acad´ emie des Sc iences 343 (2006), 679–684

2006
[22]

J. M. Lasry and P. L. Lions, Mean ﬁeld games , Japanese Journal of Mathematics 2 (2007), 229–260

2007
[23]

Liu and J

Z. Liu and J. Ma, Existence, uniqueness and exponential ergodicity under Ly apunov con- ditions for McKean-Vlasov SDEs with Markovian switching , Journal of Diﬀerential Equa- tions 337 (2022), 138–167

2022
[24]

Liu and J

Z. Liu and J. Ma, Existence, uniqueness and ergodicity for McKean-Vlasov SD Es under distribution-dependent Lyapunov conditions , (2023), arXiv:2309.05411

2023 arXiv
[25]

H. P. McKean, A class of Markov processes associated with nonlinear parab olic equations, Proceedings of the National Academy of Sciences of the United Sta tes of America 56 (1966), 1907–1911

1966
[26]

Poincar´ e,M´ emoire sur les courbes d´ eﬁnies par une ´ equation diﬀ´ erentielle I , Journal de Math´ ematiques Pures et Appliqu´ ees7 (1881), 375–422

H. Poincar´ e,M´ emoire sur les courbes d´ eﬁnies par une ´ equation diﬀ´ erentielle I , Journal de Math´ ematiques Pures et Appliqu´ ees7 (1881), 375–422
[27]

Poincar´ e, M´ emoire sur les courbes d´ eﬁnies par une ´ equation diﬀ´ erentielle II , Journal de Math´ ematiques Pures et Appliqu´ ees8 (1882), 251–296

H. Poincar´ e, M´ emoire sur les courbes d´ eﬁnies par une ´ equation diﬀ´ erentielle II , Journal de Math´ ematiques Pures et Appliqu´ ees8 (1882), 251–296
[28]

Poincar´ e,M´ emoire sur les courbes d´ eﬁnies par une ´ equation diﬀ´ erentielle III , Journal de Math´ ematiques Pures et Appliqu´ ees1 (1885), 167–244

H. Poincar´ e,M´ emoire sur les courbes d´ eﬁnies par une ´ equation diﬀ´ erentielle III , Journal de Math´ ematiques Pures et Appliqu´ ees1 (1885), 167–244
[29]

Poincar´ e,M´ emoire sur les courbes d´ eﬁnies par une ´ equation diﬀ´ erentielle IV , Journal de Math´ ematiques Pures et Appliqu´ ees2 (1886), 151–218

H. Poincar´ e,M´ emoire sur les courbes d´ eﬁnies par une ´ equation diﬀ´ erentielle IV , Journal de Math´ ematiques Pures et Appliqu´ ees2 (1886), 151–218. 20
[30]

P. Ren, M. R¨ ockner, and F. Y. Wang, Linearization of nonlinear Fokker-Planck equations and applications , Journal of Diﬀerential Equations 322 (2022), 1–37

2022
[31]

P. V. Subrahmanyam, Elementary Fixed Point Theorems , Springer, 2018

2018
[32]

A. S. Sznitman, Topics in propagation of chaos , ´Ecole d’´Et´ e de Probabilit´ es de Saint-Flour XIX-1989 (1991), 165–251

1991
[33]

Tugaut, Phase transitions of McKean-Vlasov processes in double-we lls landscape , Stochastics 86 (2014), 257–284

J. Tugaut, Phase transitions of McKean-Vlasov processes in double-we lls landscape , Stochastics 86 (2014), 257–284

2014
[34]

A. Y. Veretennikov, On ergodic measures for McKean-Vlasov stochastic equation s, Monte Carlo and Quasi-Monte Carlo Methods 2004 (2006), 470–486

2006
[35]

Villani, Optimal Transport: Old and New , Springer, 2016

C. Villani, Optimal Transport: Old and New , Springer, 2016

2016
[36]

F. Y. Wang, Distribution dependent SDEs for Landau type equations , Stochastic Processes and Their Applications 128 (2018), 595–621

2018
[37]

D. Xu, Y. Huang, and Z. Yang, Existence theorems for periodic Markov process and stochas- tic functional diﬀerential equations , Discrete and Continuous Dynamical Systems-Series A 24 (2009), 1005–1023

2009
[38]

S. Q. Zhang, Existence and non-uniqueness of stationary distributions for distribution dependent SDEs, Electronic Journal of Probability 28 (2023), 1–34

2023
[39]

Zhang, K

X. Zhang, K. Wang, and D. Li, Stochastic periodic solutions of stochastic diﬀerential equations driven by L´ evy process, Journal of Mathematical Analysis and Applications 430 (2015), 231–242. 21

2015

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [1]

N. U. Ahmed and X. Ding, On invariant measures of nonlinear Markov processes , Journal of Applied Mathematics and Stochastic Analysis 6 (1993), 385–406

work page 1993

[2] [2]

J. Bao, G. D. Reis, and Y. Wu, The random periodic solutions for McKean-Vlasov stochas- tic diﬀerential equations , (2024), arXiv:2408.17242

arXiv 2024

[3] [3]

J. Bao, M. Scheutzow, and C. Yuan, Existence of invariant probability measures for func- tional McKean-Vlasov SDEs , Electronic Journal of Probability 27 (2022), 1–14

work page 2022

[4] [4]

V. I. Bogachev, M. R¨ ockner, and S. V. Shaposhnikov, Convergence in variation of solu- tions of nonlinear Fokker-Planck-Kolmogorov equations to stationary measures, Journal of Functional Analysis 276 (2019), 3681–3713

work page 2019

[5] [5]

O. A. Butkovsky, On ergodic properties of nonlinear Markov chains and stocha stic McKean-Vlasov equations , Theory of Probability and Its Applications 58 (2014), 661– 674

work page 2014

[6] [6]

Carmona and F

R. Carmona and F. Delarue, Probabilistic Theory of Mean Field Games with Applications , I and II , Springer, 2018

work page 2018

[7] [7]

F. Chen, Y. Han, Y. Li, and X. Yang, Periodic solutions of Fokker-Planck equations , Journal of Diﬀerential Equations 263 (2017), 285–298

work page 2017

[8] [8]

D. A. Dawson, Critical dynamics and ﬂuctuations for a mean ﬁeld model of co operative behaviour, Journal of Statistical Physics 41 (1983), 29–85

work page 1983