pith. sign in

arxiv: 2507.12411 · v3 · submitted 2025-07-16 · 🧮 math.OC · cs.NA· math-ph· math.MP· math.NA

Linearization-Based Feedback Stabilization of McKean-Vlasov PDEs

Dante Kalise , Lucas M. Moschen , Grigorios A. Pavliotis This is my paper

Pith reviewed 2026-05-19 04:22 UTC · model grok-4.3

classification 🧮 math.OC cs.NAmath-phmath.MPmath.NA
keywords McKean-Vlasov PDEfeedback stabilizationRiccati feedbackground-state transformHautus testlocal exponential stabilitymaximal regularitymean-field control
0
0 comments X

The pith

A Riccati feedback law stabilizes McKean-Vlasov PDEs locally exponentially around a target stationary distribution.

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

The paper establishes a feedback control strategy for McKean-Vlasov PDEs that can steer the probability distribution of interacting particles toward a desired stationary state or hasten its convergence. This is achieved by reformulating the controlled dynamics in a suitable function space, applying a ground-state transformation to facilitate spectral analysis, and deriving a feedback law from the solution of a Riccati equation for the linearized system. A sympathetic reader would care because such PDEs arise in models of collective behavior in physics and social sciences, where controlling synchronization or stability has direct applications. The result provides a rigorous proof of local exponential stabilization with an adjustable rate using tools from infinite-dimensional systems theory.

Core claim

The authors construct a stabilizing feedback operator from the solution of an algebraic Riccati equation associated to the linearized McKean-Vlasov operator after a ground-state transformation. This feedback, when applied to the full nonlinear PDE, yields local exponential convergence to the equilibrium in a suitable norm, with the rate determined by the choice of control parameters. The construction relies on spectral analysis and verification of the infinite-dimensional Hautus test for the transformed operator.

What carries the argument

The Riccati feedback law derived from the spectral analysis of the ground-state transformed linearized operator, which provides the stabilizing control input via a time-dependent potential.

If this is right

  • Local exponential stabilization is achieved with a user-specified convergence rate.
  • The strategy applies to the noisy Kuramoto model, O(2) spin model, and von Mises potentials as demonstrated numerically.
  • Unstable equilibria can be stabilized and convergence to stationary distributions can be accelerated.
  • The control is realized through a time-dependent potential acting on the torus.

Where Pith is reading between the lines

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

  • The framework may extend to other mean-field models beyond the torus domain.
  • Prescribing the rate could enable adaptive control in real-time applications of interacting systems.
  • Further analysis might reveal how the method performs under parameter variations in the interaction kernel.

Load-bearing premise

The linearized operator around the target equilibrium must satisfy the infinite-dimensional Hautus test to permit construction of a stabilizing Riccati feedback.

What would settle it

A numerical simulation or analytical counterexample showing that the closed-loop system does not exhibit the predicted exponential decay rate for initial conditions near the equilibrium would falsify the stabilization claim.

Figures

Figures reproduced from arXiv: 2507.12411 by Dante Kalise, Grigorios A. Pavliotis, Lucas M. Moschen.

Figure 1
Figure 1. Figure 1: presents the distribution of ¯µ for K = 0.8, 1.2, 2.0 and 3.0. 0 π 2 π 3π 2 2π θ 0.0 0.2 0.4 0.6 0.8 ¯µ(θ) K = 0.8 K = 1.2 K = 2.0 K = 3.0 [PITH_FULL_IMAGE:figures/full_fig_p016_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Spectral Gap Estimates in the Noisy Kuramoto Model. Scatter plot of the lower bound for the spectral gap (in blue [13]) and the computed spectral gap (black) as functions of the coupling parameter K. 1. Acceleration to the (stable) incoherent state. For K < 1, the uniform density µ¯ = (2π) −1 is the unique stable equilibrium. Starting from a small non-uniform perturbation, Figure (a) compares the uncontrol… view at source ↗
Figure 3
Figure 3. Figure 3: Results for the control for Noisy Kuramoto Model. (a) Acceleration to the incoherent state (K = 0.95): Riccati feedback (blue) vs. uncontrolled (red dashed). (b) Stabilization of the unstable uniform state (K = 5): feedback recovers exponential decay. (c) Steering under a different stable steady state (different translation) (K = 5): feedback still maintains decay. All panels plot ∥y(t) − µ¯∥ on a log-scal… view at source ↗
Figure 4
Figure 4. Figure 4: compares the time-evolution of the probability density with and without feedback. In the uncontrolled case (left) an initially nearly uniform distribution (black) drifts toward a strongly synchronized profile (shades of gray) and finally concentrates at a clustered state (red). By contrast, under our Riccati feedback (right), the density remains close to the uniform target: despite the same initial perturb… view at source ↗
Figure 5
Figure 5. Figure 5 [PITH_FULL_IMAGE:figures/full_fig_p019_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Heatmap of Convergence for cosine-potential perturbation. Heatmaps of the base￾10 logarithm of the L 2 -norm ∥y(t)∥ versus time t (horizontal axis) and temperature σ (vertical axis) for the confining potential V (x) = 0.05 cos(2x) and interaction W(x) = − cos x. Top: controlled dynamics under the Riccati feedback law. Bottom: uncontrolled dynamics. White contour lines indicate levels log10 ∥y∥ ∈ {−8, −6, −… view at source ↗
Figure 7
Figure 7. Figure 7: Control for the O(2) model. Decay of the perturbation norm ∥y(t)∥ for the O(2) model. Left: controlled dynamics under Riccati feedback. Right: uncontrolled dynamics. 4.3 Attracitve Von Mises interaction Consider the interaction term W(x) = −I0(θ) −1 exp θ cos x  , x ∈ [0, 2π], 20 [PITH_FULL_IMAGE:figures/full_fig_p020_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Control for the Von Mises model. Comparison of the decay of the perturbation norm ∥y(·, t)∥ (log scale) under Riccati feedback (left) versus without control (right) for the Von Mises model. Feedback uniformly accelerates convergence across all σ, whereas the uncontrolled dynamics only decay rapidly when σ is large. In all cases the Riccati-based feedback yields a clear acceleration of decay and can even st… view at source ↗
read the original abstract

We develop a feedback control framework for stabilizing the McKean-Vlasov PDE on the torus. Our goal is to steer the dynamics toward a prescribed stationary distribution or accelerate convergence to it using a time-dependent control potential. We reformulate the controlled PDE in a weighted, zero-mean space and apply the ground-state transform to obtain a Schrodinger-type operator. The resulting operator framework enables spectral analysis, verification of the infinite-dimensional Hautus test, and construction of a Riccati-based feedback law derived from the linearized dynamics, yielding local exponential stabilization with a chosen convergence rate. We rigorously prove local exponential stabilization via maximal regularity arguments and nonlinear estimates. Numerical experiments on well-studied models in one and two dimensions (the noisy Kuramoto model for synchronization, the O(2) spin model in a magnetic field, and the von Mises attractive interaction potential) showcase the effectiveness of our control strategy, demonstrating convergence acceleration and stabilization of unstable equilibria.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper develops a feedback stabilization framework for controlled McKean-Vlasov PDEs on the torus. It reformulates the system in a weighted zero-mean space, applies the ground-state transform to obtain a Schrödinger-type operator, performs spectral analysis to verify the infinite-dimensional Hautus test, constructs a Riccati-based feedback law from the linearized dynamics, and proves local exponential stabilization with a prescribed rate via maximal regularity and nonlinear perturbation estimates. Numerical illustrations are provided for the noisy Kuramoto model, O(2) spin model, and von Mises interaction potential.

Significance. If the central claims hold, the work supplies a systematic linearization-based method for local stabilization of nonlocal mean-field PDEs with explicit convergence rate, extending Riccati theory and maximal regularity techniques to this setting. The combination of ground-state transform, Hautus verification, and rigorous nonlinear estimates is a technical strength; the numerical examples on standard synchronization models further support applicability to interacting particle systems.

major comments (2)
  1. [§3] §3 (Spectral analysis and Hautus test): The verification that the linearized operator (after ground-state transform and reformulation in the weighted zero-mean space) satisfies the infinite-dimensional Hautus test is asserted to enable the algebraic Riccati equation and stabilizing feedback, but the precise range condition (unstable eigenvalues lying in the range of the control operator) is not shown explicitly for the target equilibria or kernels; this check is load-bearing for the existence of the feedback operator and must be uniform in the prescribed rate.
  2. [§4] §4 (Feedback construction): The domain of the resulting Riccati feedback operator is not specified with sufficient precision (e.g., relative to the maximal regularity space), which is required to close the nonlinear estimates in the subsequent local stabilization proof.
minor comments (2)
  1. [§2] The notation distinguishing the original McKean-Vlasov variable from the transformed Schrödinger variable could be made more consistent across sections to improve readability.
  2. [§4] A brief remark on how the chosen convergence rate enters the Riccati equation would clarify the dependence of the feedback gain on this parameter.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough reading and constructive comments on our manuscript. The suggestions help clarify key technical points in the spectral analysis and feedback construction. We address each major comment below and will revise the manuscript to incorporate the requested details while preserving the core results on linearization-based stabilization of McKean-Vlasov PDEs.

read point-by-point responses

  1. Referee: §3 (Spectral analysis and Hautus test): The verification that the linearized operator (after ground-state transform and reformulation in the weighted zero-mean space) satisfies the infinite-dimensional Hautus test is asserted to enable the algebraic Riccati equation and stabilizing feedback, but the precise range condition (unstable eigenvalues lying in the range of the control operator) is not shown explicitly for the target equilibria or kernels; this check is load-bearing for the existence of the feedback operator and must be uniform in the prescribed rate.

    Authors: We agree that an explicit verification of the range condition strengthens the argument. In Section 3, the infinite-dimensional Hautus test is established via the spectral decomposition of the ground-state transformed operator, showing that the unstable spectrum is finite and that the control operator acts nontrivially on the corresponding eigenspaces for the chosen kernels. To address the referee's point directly, we will add a new lemma in the revised Section 3 that explicitly confirms, for each target equilibrium (uniform distribution and synchronized states) and admissible control potentials, that every unstable eigenvalue lies in the range of the control operator. The argument will be shown to hold uniformly for stabilization rates below the spectral gap of the linearized operator, using the explicit form of the eigenfunctions on the torus. This addition closes the gap without altering the existing proofs. revision: yes

  2. Referee: §4 (Feedback construction): The domain of the resulting Riccati feedback operator is not specified with sufficient precision (e.g., relative to the maximal regularity space), which is required to close the nonlinear estimates in the subsequent local stabilization proof.

    Authors: We acknowledge that greater precision on the domain is needed for rigor. The Riccati feedback is obtained from the solution of the algebraic Riccati equation associated with the linearized operator in the weighted zero-mean space. In the revised manuscript we will explicitly define the domain of the feedback operator as the intersection of the domain of the linearized generator with the maximal regularity space (specifically, functions whose time derivative lies in L^2(0,T; H^{-1}) and whose spatial regularity satisfies the appropriate Sobolev embedding). We will also insert a short proposition verifying that this domain is invariant under the nonlinear perturbation and that the feedback term maps into the control space, thereby allowing the nonlinear estimates in the local stabilization theorem to close directly. These clarifications will be added to Section 4 without changing the overall proof strategy. revision: yes

Circularity Check

0 steps flagged

No circularity: standard linear control theory applied to transformed operator with explicit verification steps

full rationale

The derivation begins with reformulation of the controlled McKean-Vlasov PDE into a weighted zero-mean space, applies the ground-state transform to obtain a Schrödinger-type operator, performs spectral analysis, verifies the infinite-dimensional Hautus test for the linearized operator, constructs the Riccati feedback, and closes the local exponential stabilization via maximal regularity and nonlinear perturbation estimates. Each step invokes external functional-analytic tools (maximal regularity, Riccati theory) rather than reducing the target stabilization claim to a fitted parameter, self-referential definition, or unverified self-citation chain. The Hautus test is treated as a model-dependent condition to be checked after spectral analysis, not smuggled in as an ansatz or renamed known result; the nonlinear estimates are independent of the linear feedback construction. No load-bearing premise collapses to the paper's own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The abstract invokes standard functional-analysis results (maximal regularity for parabolic operators) and control-theoretic lemmas (infinite-dimensional Hautus test) without introducing new free parameters or invented entities.

axioms (2)
  • domain assumption The linearized McKean-Vlasov operator satisfies the infinite-dimensional Hautus test after the ground-state transform.
    Invoked to guarantee existence of the Riccati feedback law.
  • standard math Maximal regularity holds for the controlled parabolic operator on the torus.
    Used to obtain the nonlinear estimates that close the local stabilization proof.

pith-pipeline@v0.9.0 · 5711 in / 1420 out tokens · 29301 ms · 2026-05-19T04:22:39.851351+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

46 extracted references · 46 canonical work pages

  1. [1]

    J. A. Acebr´ on, L. L. Bonilla, Conrad J. P´ erez V., F. Ritort, and R. Spigler. The kuramoto model: A simple paradigm for synchronization phenomena. Rev. Mod. Phys., 77:137–185, Apr 2005

    work page 2005

  2. [2]

    Control of high-dimensional collective dynamics by deep neural feedback laws and kinetic modelling

    Giacomo Albi, Sara Bicego, and Dante Kalise. Control of high-dimensional collective dynamics by deep neural feedback laws and kinetic modelling. Journal of Computational Physics , page 114229, 2025

    work page 2025

  3. [3]

    Mean field control hierarchy

    Giacomo Albi, Young-Pil Choi, Massimo Fornasier, and Dante Kalise. Mean field control hierarchy. Applied Mathematics & Optimization , 76(1):93–135, 2017

    work page 2017

  4. [4]

    Springer, 2005

    Luigi Ambrosio, Nicola Gigli, and Giuseppe Savar´ e.Gradient flows: in metric spaces and in the space of probability measures. Springer, 2005

    work page 2005

  5. [5]

    A maximum principle for SDEs of mean-field type

    Daniel Andersson and Boualem Djehiche. A maximum principle for SDEs of mean-field type. Applied Mathematics & Optimization , 63(3):341–356, 2011

    work page 2011

  6. [6]

    Well-posedness and stationary solutions of McKean-Vlasov (S) PDEs

    Letizia Angeli, Julien Barr´ e, Martin Kolodziejczyk, and Michela Ottobre. Well-posedness and stationary solutions of McKean-Vlasov (S) PDEs. Journal of Mathematical Analysis and Applications, 526(2):127301, 2023

    work page 2023

  7. [7]

    A Fokker–Planck control framework for stochastic systems

    Mario Annunziato and Alfio Borz` ı. A Fokker–Planck control framework for stochastic systems. EMS Surveys in Mathematical Sciences , 5(1):65–98, 2018

    work page 2018

  8. [8]

    Soledad Aronna and F

    M. Soledad Aronna and F. Tr¨ oltzsch. First and second order optimality conditions for the control of Fokker-Planck equations. ESAIM: COCV, 27:15, 2021

    work page 2021

  9. [9]

    Exact controllability of Fokker–Planck equations and McKean–Vlasov SDEs

    Viorel Barbu. Exact controllability of Fokker–Planck equations and McKean–Vlasov SDEs. SIAM Journal on Control and Optimization , 61(3):1805–1818, 2023

    work page 2023

  10. [10]

    Becker and A

    S. Becker and A. Menegaki. Spectral gap in mean-field O(n)-model. Comm. Math. Phys. , 380(3):1361–1400, 2020

    work page 2020

  11. [11]

    Representation and control of infinite dimensional systems

    Alain Bensoussan, Giuseppe Da Prato, Michel C Delfour, and Sanjoy K Mitter. Representation and control of infinite dimensional systems . Springer, 2007

    work page 2007

  12. [12]

    Mean Field Games and Mean Field Type Control

    Alain Bensoussan, Jens Frehse, and Philip Yam. Mean Field Games and Mean Field Type Control. Springer, New York, 2013

    work page 2013

  13. [13]

    Dynamical aspects of mean field plane rotators and the Kuramoto model

    Lorenzo Bertini, Giambattista Giacomin, and Khashayar Pakdaman. Dynamical aspects of mean field plane rotators and the Kuramoto model. Journal of Statistical Physics , 138:270–290, 2010. 22

    work page 2010

  14. [14]

    Stability of stationary states for mean field models with multichromatic interaction potentials

    Benedetta Bertoli, Benjamin D Goddard, and Grigorios A Pavliotis. Stability of stationary states for mean field models with multichromatic interaction potentials. IMA Journal of Applied Mathematics, 89(5):833–859, 2024

    work page 2024

  15. [15]

    Bicego, D

    S. Bicego, D. Kalise, and G. A. Pavliotis. Computation and control of unstable steady states for mean field multiagent systems. arXiv, 2024

    work page 2024

  16. [16]

    Controllability of Fokker-Planck equations, the Schr¨ odinger system under related stochastic optimal control

    Andr´ e Blaqui` ere. Controllability of Fokker-Planck equations, the Schr¨ odinger system under related stochastic optimal control. Dynamics and Control , 2(3):235–253, 1992

    work page 1992

  17. [17]

    Mean-field Pontryagin maximum principle

    Mattia Bongini, Massimo Fornasier, Francesco Rossi, and Francesco Solombrino. Mean-field Pontryagin maximum principle. Journal of Optimization Theory and Applications , 175(1):1–38, 2017

    work page 2017

  18. [18]

    Breiten, K

    T. Breiten, K. Kunisch, and L. Pfeiffer. Control strategies for the Fokker-Planck equation. ESAIM: COCV, 24(2), 2018

    work page 2018

  19. [19]

    Breiten and K

    T. Breiten and K. K. Kunisch. Improving the convergence rates for the kinetic Fokker-Planck equation by optimal control. SIAM J. Control Optim. , 61(3):1557–1581, 2023

    work page 2023

  20. [20]

    Springer, 2018

    Ren´ e Carmona, Fran¸ cois Delarue, et al.Probabilistic theory of mean field games with applications I-II, volume 3. Springer, 2018

    work page 2018

  21. [21]

    Control of McKean-Vlasov dynamics versus mean field games

    Ren´ e Carmona, Fran¸ cois Delarue, and Aim´ e Lachapelle. Control of McKean-Vlasov dynamics versus mean field games. Mathematics and Financial Economics , 7:131–166, 2013

    work page 2013

  22. [22]

    J. A. Carrillo, R. S. Gvalani, G. A. Pavliotis, and A. Schlichting. Long-time behaviour and phase transitions for the McKean-Vlasov equation on the torus. Archive for Rational Mechanics and Analysis, 235:635–690, 2020

    work page 2020

  23. [23]

    Carrillo, Robert J

    Jos´ e A. Carrillo, Robert J. McCann, and C´ edric Villani. Kinetic equilibration rates for granular media and related equations: entropy dissipation and mass transportation estimates. Revista Matem´ atica Iberoamericana, 19(3):971–1018, 2003

    work page 2003

  24. [24]

    The McKean–Vlasov equation in finite volume

    Lina Chayes and Vladimir Panferov. The McKean–Vlasov equation in finite volume. Journal of Statistical Physics , 138(1–3):351–380, 2010

    work page 2010

  25. [25]

    Optimal control for Kuramoto model: from many-particle liouville equation to diffusive mean-field problem

    Li Chen, Yucheng Wang, and Valeriia Zhidkova. Optimal control for Kuramoto model: from many-particle liouville equation to diffusive mean-field problem. arXiv preprint arXiv:2504.09425, 2025

    work page arXiv 2025

  26. [26]

    Control and nonlinearity

    Jean-Michel Coron. Control and nonlinearity. Number 136 in 1. American Mathematical Soc., 2007

    work page 2007

  27. [27]

    An introduction to infinite-dimensional linear systems theory , volume 21

    Ruth F Curtain and Hans Zwart. An introduction to infinite-dimensional linear systems theory , volume 21. Springer Science & Business Media, 2012

    work page 2012

  28. [28]

    Donald A. Dawson. Critical dynamics and fluctuations for a mean-field model of cooperative behavior. Journal of Statistical Physics , 31(1):29–85, 1983

    work page 1983

  29. [29]

    Fleig and R

    A. Fleig and R. Guglielmi. Optimal control of the Fokker-Planck equation with space-dependent controls. Journal of Optimization Theory and Applications , 174, Aug. 2017. 23

    work page 2017

  30. [30]

    Garnier, G

    J. Garnier, G. Papanicolaou, and T.-W. Yang. Consensus convergence with stochastic effects. Vietnam J. Math. , 45(1-2):51–75, 2017

    work page 2017

  31. [31]

    Opinion dynamics and bounded confidence models, analysis, and simulation

    Rainer Hegselmann and Ulrich Krause. Opinion dynamics and bounded confidence models, analysis, and simulation. Journal of Artifical Societies and Social Simulation (JASSS) vol , 5(3), 2002

    work page 2002

  32. [32]

    Malham´ e, and Peter E

    Minyi Huang, Roland P. Malham´ e, and Peter E. Caines. Large population stochastic dynamic games: closed-loop McKean-Vlasov systems and the Nash certainty equivalence principle. Communications in Information & Systems , 6(3):221 – 252, 2006

    work page 2006

  33. [33]

    A spectral approach to optimal control of the Fokker-Planck equation

    Dante Kalise, Lucas M Moschen, Grigorios A Pavliotis, and Urbain Vaes. A spectral approach to optimal control of the Fokker-Planck equation. IEEE Control Systems Letters , 2025

    work page 2025

  34. [34]

    Perturbation theory for linear operators , volume 132

    Tosio Kato. Perturbation theory for linear operators , volume 132. Springer Science & Business Media, 2013

    work page 2013

  35. [35]

    Mean field games

    Jean-Michel Lasry and Pierre-Louis Lions. Mean field games. Japanese journal of mathematics , 2(1):229–260, 2007

    work page 2007

  36. [36]

    Non-homogeneous boundary value problems and applications: Vol

    Jacques Louis Lions and Enrico Magenes. Non-homogeneous boundary value problems and applications: Vol. 1 , volume 181. Springer Science & Business Media, 2012

    work page 2012

  37. [37]

    Logarithmic Sobolev inequalities for some nonlinear PDE’s

    Florient Malrieu. Logarithmic Sobolev inequalities for some nonlinear PDE’s. Stochastic processes and their applications, 95(1):109–132, 2001

    work page 2001

  38. [38]

    H. P. McKean. Propagation of chaos for a class of non-linear parabolic equations. Lecture Series in Differential Equations, Volume 2 (ed. A. K. Aziz), no. 19 in Van Nostrand Mathematical Studies, pages 177–194, 1969

    work page 1969

  39. [39]

    A class of Markov processes associated with nonlinear parabolic equations

    Henry P McKean Jr. A class of Markov processes associated with nonlinear parabolic equations. Proceedings of the National Academy of Sciences , 56(6):1907–1911, 1966

    work page 1907

  40. [40]

    Bayesian nonparametric inference in McKean-Vlasov models

    Richard Nickl, Grigorios A Pavliotis, and Kolyan Ray. Bayesian nonparametric inference in McKean-Vlasov models. The Annals of statistics , 53(1):170–193, 2025

    work page 2025

  41. [41]

    A martingale approach to the law of large numbers for weakly interacting stochastic processes

    Klaus Oelschl¨ ager. A martingale approach to the law of large numbers for weakly interacting stochastic processes. Annals of Probability, 12(2):458–479, 1984

    work page 1984

  42. [42]

    Linearization of ergodic McKean SDEs and applica- tions

    Grigorios A Pavliotis and Andrea Zanoni. Linearization of ergodic McKean SDEs and applica- tions. arXiv preprint arXiv:2501.13655 , 2025

    work page arXiv 2025

  43. [43]

    Information-theoretic description of a feedback-control Kuramoto model

    Damian R Sowinski, Adam Frank, and Gourab Ghoshal. Information-theoretic description of a feedback-control Kuramoto model. Physical Review Research, 6(4):043188, 2024

    work page 2024

  44. [44]

    Stability of incoherence in a population of coupled oscillators

    Steven H Strogatz and Renato E Mirollo. Stability of incoherence in a population of coupled oscillators. Journal of Statistical Physics , 63:613–635, 1991

    work page 1991

  45. [45]

    Topics in propagation of chaos

    Alain-Sol Sznitman. Topics in propagation of chaos. Ecole d’´ et´ e de probabilit´ es de Saint-Flour XIX—1989, 1464:165–251, 1991

    work page 1989

  46. [46]

    On asymptotic behaviors of the solution of a nonlinear diffusion equation

    Yoshihisa Tamura. On asymptotic behaviors of the solution of a nonlinear diffusion equation. J. Fac. Sci. Univ. Tokyo Sect. IA Math. , 31:195–221, 1984. 24

    work page 1984