pith. sign in

arxiv: 2605.00786 · v1 · submitted 2026-05-01 · 📊 stat.ME · math.PR· stat.ML

Recursive Maximum Likelihood Estimation for Interacting Particle Systems using Virtual Particles

Louis Sharrock , Nikolas Kantas , Grigorios A. Pavliotis This is my paper

Pith reviewed 2026-05-09 18:51 UTC · model grok-4.3

classification 📊 stat.ME math.PRstat.ML
keywords recursive maximum likelihoodinteracting particle systemsmean-field limitsvirtual particlesstochastic gradientsparameter estimationcontinuous observation
0
0 comments X

The pith

Recursive maximum likelihood estimation for interacting particle systems is achieved by optimizing the mean-field stationary log-likelihood with stochastic gradients from virtual particle systems.

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

The paper develops recursive maximum likelihood estimation for stochastic interacting particle systems observed through the continuous trajectory of only one particle. Direct likelihood estimation is inconsistent for finite-particle systems even as particle number and time go to infinity, so the method instead optimizes the stationary log-likelihood of the limiting mean-field system. Stochastic gradient estimates are formed in continuous time by integrating a virtual interacting particle system and a virtual tangent particle system alongside the running parameter estimate. For fixed numbers of particles the algorithm drives the surrogate gradient to zero as time tends to infinity; in the iterated limit of infinite time followed by infinite particles the surrogate gradients converge uniformly to the mean-field gradient and the estimates converge to its stationary points. This supplies a workable route to parameter learning in large-scale systems such as neuron networks or oscillator ensembles when only sparse single-particle data are available.

Core claim

By integrating a virtual interacting particle system and virtual tangent system with the online parameter, the method produces surrogate gradients for a finite-particle objective that, for fixed N and M, are driven to zero as t tends to infinity; in the double limit t to infinity followed by N,M to infinity these gradients converge uniformly to the gradient of the stationary log-likelihood of the mean-field limit, yielding convergence of the estimates to its stationary points.

What carries the argument

The stochastic gradient estimate computed from the single observed particle trajectory together with an integrated virtual interacting particle system and virtual tangent interacting particle system.

If this is right

  • For any fixed N and M the surrogate gradient of the finite-particle objective is driven to zero as t tends to infinity.
  • In the iterated limit the surrogate gradients converge uniformly to the mean-field stationary gradient.
  • Parameter estimates therefore converge to stationary points of the mean-field objective.
  • The procedure is illustrated on quadratic confinement models, FitzHugh-Nagumo neuron networks, and the stochastic Kuramoto model.

Where Pith is reading between the lines

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

  • The same virtual-particle construction could be tested on systems whose mean-field limit is only known numerically rather than in closed form.
  • Adaptation to discrete-time or noisy partial observations would extend the method to typical experimental recordings.
  • The approach suggests that consistent parameter recovery does not require simultaneous observation of the entire interacting ensemble.
  • Links to mean-field filtering or control problems may allow reuse of the virtual tangent system for related inference tasks.

Load-bearing premise

The particle system admits a well-defined differentiable stationary log-likelihood in its mean-field limit.

What would settle it

Numerical runs with increasing N and M in which the parameter trajectory fails to approach a stationary point of the explicitly computable mean-field log-likelihood would falsify the double-limit convergence claim.

Figures

Figures reproduced from arXiv: 2605.00786 by Grigorios A. Pavliotis, Louis Sharrock, Nikolas Kantas.

Figure 1
Figure 1. Figure 1: Online parameter estimation for a model with quadratic confinement potential and quadratic interaction potential. We plot the sequence of online parameter estimates (θt)t≥0 and (ϑt)t≥0, as defined by the update equations in (10) and (11). The true parameters are given by θ0 = (1.2, 0.5)⊤. The initial parameter estimates are given by θinit,1 ∼ U(1.5, 2.5) and θinit,2 ∼ U(1.0, 1.5). 18 view at source ↗
Figure 2
Figure 2. Figure 2: Online parameter estimation for the interaction parameter in a model with quadratic confinement potential and quadratic interaction potential. We plot the sequence of online parameter estimates (θt,2)t≥0 and (ϑt,2)t≥0, as defined by the update equations in (10) and (11), for N ∈ {2, 5, 10, 100}. The true parameters are given by θ0 = (1.2, 0.5)⊤, with the first parameter assumed known. The initial parameter… view at source ↗
Figure 3
Figure 3. Figure 3: The L 2 error of the averaged and the non-averaged estimators, for a model with quadratic confinement potential and quadratic interaction potential. We plot the L 2 error for both estimators after T = 50, 000 iterations, for N ∈ {3, 5, 10, 25, 50} particles. Finally, in view at source ↗
Figure 4
Figure 4. Figure 4: The L 2 error of the averaged and the non-averaged estimators, for a model with quadratic confinement potential and quadratic interaction potential. We plot the L 2 error for both estimators after T = 50, 000 iterations, for M ∈ {5, 10, 20, 30, 40, 50} virtual particles. 19 view at source ↗
Figure 5
Figure 5. Figure 5: f). Second, for at least some of the parameters, the (asymptotic) bias of both estimators decreases 0 1000 2000 3000 4000 5000 0.00 0.25 0.50 0.75 1.00 1.25 t, 1 t, 1 t, 2 t, 2 t, 3 t, 3 (a) N = 3. 0 1000 2000 3000 4000 5000 0.00 0.25 0.50 0.75 1.00 1.25 t, 1 t, 1 t, 2 t, 2 t, 3 t, 3 (b) N = 5. 0 1000 2000 3000 4000 5000 0.00 0.25 0.50 0.75 1.00 1.25 t, 1 t, 1 t, 2 t, 2 t, 3 t, 3 (c) N = 10. 0 1000 2000 30… view at source ↗
Figure 6
Figure 6. Figure 6: Online parameter estimation for the stochastic Kuramoto model. We plot the sequence of online parameter estimates (θt)t≥0 and (ϑt)t≥0, for N ∈ {3, 10, 50}. The true time-varying parameter is given in (12). Meanwhile, the initial parameter estimate is given by θinit ∼ U[2, 3]. 21 view at source ↗
read the original abstract

We study recursive maximum likelihood estimation for stochastic interacting particle systems based on continuous observation of a single particle. In this regime, consistent estimation of the finite-particle log-likelihood is not possible, even in the limit as the number of particles $N\rightarrow\infty$ and the time horizon $t\rightarrow\infty$. We thus seek to optimise the stationary log-likelihood of the limiting mean-field system. We achieve this via a form of stochastic gradient estimate in continuous time, with stochastic gradient estimates computed using the continuous trajectory of the single observed particle, alongside a virtual interacting particle system and a virtual tangent interacting particle system, which are integrated with the online parameter estimate. For fixed numbers of real and virtual particles, we show that the resulting algorithms drive the gradient of a finite-particle surrogate objective to zero as $t\to\infty$. We then prove that, in the iterated limit $t\to\infty$ followed by $N,M\to\infty$, these surrogate gradients converge uniformly to the gradient of the stationary log-likelihood of the limiting mean-field system, yielding convergence to its stationary points. We illustrate the method on several numerical examples, including a model with quadratic confinement and interaction potentials, a model of interacting FitzHugh--Nagumo neurons, and a stochastic Kuramoto model.

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

1 major / 2 minor

Summary. The paper develops a recursive MLE procedure for stochastic interacting particle systems observed through a single continuous trajectory. It constructs online stochastic gradient estimates via an auxiliary virtual interacting particle system and a virtual tangent interacting particle system driven by the current parameter estimate. For fixed N and M the surrogate gradient is shown to converge to zero as t→∞. In the iterated limit t→∞ followed by N,M→∞ the surrogate gradients converge uniformly to the gradient of the stationary log-likelihood of the mean-field limit, implying convergence to its stationary points. The method is illustrated on a quadratic-potential model, FitzHugh–Nagumo neurons, and the stochastic Kuramoto model.

Significance. If the stated convergence results hold under verifiable conditions, the work supplies a practical online estimator for mean-field parameters from single-particle data, circumventing the inconsistency of finite-N likelihoods. The virtual-particle construction for gradient estimation is technically novel and the double-limit analysis is a substantive contribution. The approach could be useful in statistical physics and neuroscience applications where only partial observations are available.

major comments (1)
  1. [Assumptions preceding the main convergence theorem and the statements of the numerical examples] The central iterated-limit claim (abstract and main convergence theorem) requires that both the real and virtual (including tangent) particle systems possess a unique stationary measure with sufficiently rapid mixing so that time averages equal expectations and the virtual law tracks the mean-field law uniformly. The manuscript invokes ergodicity and mixing but supplies no explicit conditions on potentials, noise intensities, or coupling strengths that guarantee these properties. For the stochastic Kuramoto and FitzHugh–Nagumo examples, synchronization or oscillatory regimes can produce slow mixing or multiple invariant measures, directly threatening the uniformity needed for the double-limit interchange.
minor comments (2)
  1. [Section 2 (model and algorithm)] The distinction between the real, virtual, and tangent processes is introduced gradually; a single early diagram or table summarizing the three coupled SDEs and their driving noises would improve readability.
  2. [Numerical examples section] Numerical figures would benefit from overlaying multiple independent runs or reporting variability measures to illustrate the stochastic nature of the online estimates.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the detailed and constructive report. The positive assessment of the work's novelty and potential applications is appreciated. We address the single major comment below.

read point-by-point responses

  1. Referee: [Assumptions preceding the main convergence theorem and the statements of the numerical examples] The central iterated-limit claim (abstract and main convergence theorem) requires that both the real and virtual (including tangent) particle systems possess a unique stationary measure with sufficiently rapid mixing so that time averages equal expectations and the virtual law tracks the mean-field law uniformly. The manuscript invokes ergodicity and mixing but supplies no explicit conditions on potentials, noise intensities, or coupling strengths that guarantee these properties. For the stochastic Kuramoto and FitzHugh–Nagumo examples, synchronization or oscillatory regimes can produce slow mixing or multiple invariant measures, directly threatening the uniformity needed for the double-limit interchange.

    Authors: We agree that the manuscript would benefit from greater explicitness regarding the ergodicity and mixing assumptions that underpin the convergence statements. In the revised version we will insert a dedicated subsection (immediately preceding the main theorem) that states verifiable sufficient conditions: Lipschitz continuity and quadratic growth bounds on the confining and interaction potentials, an upper bound on the noise intensity relative to the potential strength, and a restriction on the coupling parameter that guarantees a positive Dobrushin coefficient (or Wasserstein contraction) for the mean-field McKean–Vlasov dynamics and its virtual/tangent counterparts. These conditions are standard in the interacting-particle literature and can be checked directly from the model coefficients. For the numerical examples we will add a short paragraph specifying the exact parameter values used and explaining why they lie inside the regime where uniqueness and exponential mixing hold (weak coupling for Kuramoto to preclude full synchronization; parameters yielding a globally attractive limit cycle for FitzHugh–Nagumo). While a exhaustive classification of every possible regime is beyond the scope of the present work, the added material will make the hypotheses checkable and will clarify the scope of the double-limit result. We view this as a partial but substantive revision that directly responds to the concern without changing the paper’s main contributions. revision: partial

Circularity Check

0 steps flagged

No circularity: convergence to external mean-field objective via auxiliary virtual systems

full rationale

The paper's derivation constructs surrogate gradients from an observed particle trajectory together with separately simulated virtual interacting particle and tangent systems that are driven by the current online parameter estimate. For fixed N and M the algorithm is shown to drive the finite-particle surrogate gradient to zero as t→∞. The subsequent iterated limit t→∞ then N,M→∞ is proved to produce uniform convergence of these surrogates to the gradient of the stationary log-likelihood of the limiting mean-field system. The target mean-field log-likelihood is an independent object defined by the mean-field limit itself; the surrogates are not defined in terms of it, nor is any fitted parameter renamed as a prediction. No self-definitional loops, load-bearing self-citations, or ansatz smuggling appear in the chain. The result is a standard stochastic-approximation convergence theorem resting on external ergodicity and mixing assumptions rather than on any reduction to its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 2 invented entities

The approach rests on the existence of a well-behaved mean-field limit and ergodicity of the limiting process; virtual systems are computational devices rather than new physical postulates. No free parameters are fitted in the theoretical claims.

axioms (1)
  • domain assumption The finite-particle system converges to a well-defined mean-field limit whose stationary log-likelihood is differentiable and has isolated stationary points.
    Invoked to justify targeting the mean-field objective instead of the finite-N likelihood.
invented entities (2)
  • virtual interacting particle system no independent evidence
    purpose: To generate online stochastic gradient estimates from the single observed trajectory
    Computational auxiliary system integrated with the parameter estimate; no independent physical evidence provided.
  • virtual tangent interacting particle system no independent evidence
    purpose: To compute derivatives of the surrogate objective with respect to parameters
    Auxiliary system for gradient computation; introduced as part of the algorithm.

pith-pipeline@v0.9.0 · 5527 in / 1312 out tokens · 37231 ms · 2026-05-09T18:51:02.864219+00:00 · methodology

discussion (0)

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

Reference graph

Works this paper leans on

92 extracted references · 92 canonical work pages

  1. [1]

    A., Bonilla, L

    Acebr \'o n, J. A., Bonilla, L. L., P \'e rez Vicente, C. J., Ritort, F., and Spigler, R. (2005). The Kuramoto model: A simple paradigm for synchronization phenomena. Reviews of Modern Physics , 77(1):137--185

    work page 2005

  2. [2]

    Amorino, C., Belomestny, D., Pilipauskait \.e , V., Podolskij, M., and Zhou, S.-Y. (2025). Polynomial rates via deconvolution for nonparametric estimation in McKean--Vlasov SDEs . Probability Theory and Related Fields , 193:539--584

    work page 2025

  3. [3]

    Amorino, C., Heidari, A., Pilipauskait \.e , V., and Podolskij, M. (2023). Parameter estimation of discretely observed interacting particle systems. Stochastic Processes and their Applications , 163:350--386

    work page 2023

  4. [4]

    Amorino, V

    Amorino, C. and Pilipauskait \.e , V. (2024). Kinetic interacting particle system: parameter estimation from complete and partial discrete observations. arXiv preprint arXiv:2410.10226

    work page arXiv 2024

  5. [5]

    Baladron, J., Fasoli, D., Faugeras, O., and Touboul, J. (2012). Mean-field description and propagation of chaos in networks of Hodgkin--Huxley and FitzHugh--Nagumo neurons. Journal of Mathematical Neuroscience , 2(1):10

    work page 2012

  6. [6]

    Bashiri, K. (2020). On the long-time behaviour of McKean--Vlasov paths. Electronic Communications in Probability , 25:1--14

    work page 2020

  7. [7]

    Bauer, M., Meyer-Brandis, T., and Proske, F. (2018). Strong solutions of mean-field stochastic differential equations with irregular drift. Electronic Journal of Probability , 23:1--35

    work page 2018

  8. [8]

    Belomestny, D., Podolskij, M., and Zhou, S.-Y. (2024). On nonparametric estimation of the interaction function in particle system models. arXiv preprint arXiv:2402.14419

    work page arXiv 2024

  9. [9]

    Benachour, S., Roynette, B., Talay, D., and Vallois, P. (1998). Nonlinear self-stabilizing processes I : Existence, invariant probability, propagation of chaos. Stochastic Processes and their Applications , 75(2):173--201

    work page 1998

  10. [10]

    Benedetto, D., Caglioti, E., and Pulvirenti, M. (1997). A kinetic equation for granular media. Mathematical Modelling and Numerical Analysis , 31(5):615--641

    work page 1997

  11. [11]

    Bertini, L., Giacomin, G., and Pakdaman, K. (2010). Dynamical aspects of mean field plane rotators and the Kuramoto model. Journal of Statistical Physics , 138(1--3):270--290

    work page 2010

  12. [12]

    and Cartea, \'A

    Bhudisaksang, T. and Cartea, \'A . (2021). Online drift estimation for jump-diffusion processes. Bernoulli , 27(4):2494--2518

    work page 2021

  13. [13]

    Bishwal, J. P. N. (2011). Estimation in interacting diffusions: Continuous and discrete sampling. Applied Mathematics , 2(9):1154--1158

    work page 2011

  14. [14]

    Bolley, F., Gentil, I., and Guillin, A. (2013). Uniform convergence to equilibrium for granular media. Archive for Rational Mechanics and Analysis , 208(2):429--445

    work page 2013

  15. [15]

    S., and Spiliopoulos, K

    Bourguin, S., Dhama, S. S., and Spiliopoulos, K. (2026). Quantitative fluctuation analysis for continuous-time stochastic gradient descent via Malliavin calculus. arXiv preprint arXiv:2603.07149

    work page arXiv 2026

  16. [16]

    Buckdahn, R., Li, J., and Ma, J. (2017). A mean-field stochastic control problem with partial observations. The Annals of Applied Probability , 27(5):3201--3245

    work page 2017

  17. [17]

    Burger, M., Capasso, V., and Morale, D. (2007). On an aggregation model with long and short range interactions. Nonlinear Analysis: Real World Applications , 8(3):939--958

    work page 2007

  18. [18]

    Canuto, C., Fagnani, F., and Tilli, P. (2012). An Eulerian approach to the analysis of Krause 's consensus models. SIAM Journal on Control and Optimization , 50(1):243--265

    work page 2012

  19. [19]

    Cardaliaguet, P., Delarue, F., Lasry, J.-M., and Lions, P.-L. (2019). The Master Equation and the Convergence Problem in Mean Field Games , volume 201 of Annals of Mathematics Studies . Princeton University Press, Princeton, NJ

    work page 2019

  20. [20]

    and Lehalle, C.-A

    Cardaliaguet, P. and Lehalle, C.-A. (2018). Mean field game of controls and an application to trade crowding. Mathematics and Financial Economics , 12(3):335--363

    work page 2018

  21. [21]

    and Delarue, F

    Carmona, R. and Delarue, F. (2018). Probabilistic Theory of Mean Field Games with Applications I . Springer-Verlag, Cham, Switzerland

    work page 2018

  22. [22]

    A., Gvalani, R

    Carrillo, J. A., Gvalani, R. S., Pavliotis, G. A., and Schlichting, A. (2020). Long-time behaviour and phase transitions for the McKean--Vlasov equation on the torus. Archive for Rational Mechanics and Analysis , 235(1):635--690

    work page 2020

  23. [23]

    A., McCann, R

    Carrillo, J. A., McCann, R. J., and Villani, C. (2006). Contractions in the 2- Wasserstein length space and thermalization of granular media. Archive for Rational Mechanics and Analysis , 179(2):217--263

    work page 2006

  24. [24]

    Cattiaux, P., Guillin, A., and Malrieu, F. (2008). Probabilistic approach for granular media equations in the non-uniformly convex case. Probability Theory and Related Fields , 140(1--2):19--40

    work page 2008

  25. [25]

    and Diez, A

    Chaintron, L.-P. and Diez, A. (2022a). Propagation of chaos: A review of models, methods and applications. I . models and methods. Kinetic and Related Models , 15(6):895--1015

    work page

  26. [26]

    and Diez, A

    Chaintron, L.-P. and Diez, A. (2022b). Propagation of chaos: A review of models, methods and applications. II . applications. Kinetic and Related Models , 15(6):1017--1173

    work page

  27. [27]

    Chaudru de Raynal, P.-E. (2020). Strong well posedness of McKean--Vlasov stochastic differential equations with H \"o lder drift. Stochastic Processes and their Applications , 130(1):79--107

    work page 2020

  28. [28]

    Chazelle, B., Jiu, Q., Li, Q., and Wang, C. (2017). Well-posedness of the limiting equation of a noisy consensus model in opinion dynamics. Journal of Differential Equations , 263(1):365--397

    work page 2017

  29. [29]

    Chen, X. (2021). Maximum likelihood estimation of potential energy in interacting particle systems from single-trajectory data. Electronic Communications in Probability , 26:1--13

    work page 2021

  30. [30]

    and Genon-Catalot, V

    Comte, F. and Genon-Catalot, V. (2023). Nonparametric adaptive estimation for interacting particle systems. Scandinavian Journal of Statistics , 50(4):1716--1755

    work page 2023

  31. [31]

    Comte, F., Genon-Catalot, V., and Lar \'e do, C. (2025). Nonparametric moment method for scalar McKean--Vlasov stochastic differential equations. ESAIM: Probability and Statistics , 29:400--449

    work page 2025

  32. [32]

    and Xiong, J

    Crisan, D. and Xiong, J. (2010). Approximate McKean--Vlasov representations for a class of SPDEs . Stochastics , 82(1):53--68

    work page 2010

  33. [33]

    G., Gvalani, R

    Delgadino, M. G., Gvalani, R. S., Pavliotis, G. A., and Smith, S. A. (2023). Phase transitions, logarithmic Sobolev inequalities, and uniform-in-time propagation of chaos for weakly interacting diffusions. Communications in Mathematical Physics , 401:275--323

    work page 2023

  34. [34]

    and Hoffmann, M

    Della Maestra, L. and Hoffmann, M. (2022). Nonparametric estimation for interacting particle systems: McKean--Vlasov models. Probability Theory and Related Fields , 182(1):551--613

    work page 2022

  35. [35]

    and Hoffmann, M

    Della Maestra, L. and Hoffmann, M. (2023). The LAN property for McKean--Vlasov models in a mean-field regime. Stochastic Processes and their Applications , 155:109--146

    work page 2023

  36. [36]

    Durmus, A., Eberle, A., Guillin, A., and Zimmer, R. (2020). An elementary approach to uniform in time propagation of chaos. Proceedings of the American Mathematical Society , 148(12):5387--5398

    work page 2020

  37. [37]

    Eberle, A., Guillin, A., and Zimmer, R. (2019). Quantitative Harris -type theorems for diffusions and McKean--Vlasov processes. Transactions of the American Mathematical Society , 371(10):7135--7173

    work page 2019

  38. [38]

    and Lar \'e do, C

    Genon-Catalot, V. and Lar \'e do, C. (2024a). Inference for ergodic McKean--Vlasov stochastic differential equations with polynomial interactions. Annales de l'Institut Henri Poincar \'e (B) Probabilit \'e s et Statistiques , 60(4):2668--2693

    work page

  39. [39]

    and Lar \'e do, C

    Genon-Catalot, V. and Lar \'e do, C. (2024b). Parametric inference for ergodic McKean--Vlasov stochastic differential equations. Bernoulli , 30(3):1971--1997

    work page 1971

  40. [40]

    Gerencs \'e r, L., Gy \"o ngy, I., and Michaletzky, G. (1984). Continuous-time recursive maximum likelihood method: a new approach to Ljung 's scheme. IFAC Proceedings Volumes , 17(2):683--686

    work page 1984

  41. [41]

    and Prokaj, V

    Gerencs \'e r, L. and Prokaj, V. (2009). Recursive identification of continuous-time linear stochastic systems: convergence w.p.\ 1 and in L^q . In Proceedings of the 2009 European Control Conference (ECC) , pages 1209--1214

    work page 2009

  42. [42]

    Giesecke, K., Schwenkler, G., and Sirignano, J. A. (2020). Inference for large financial systems. Mathematical Finance , 30(1):3--46

    work page 2020

  43. [43]

    D., Gooding, B., Short, H., and Pavliotis, G

    Goddard, B. D., Gooding, B., Short, H., and Pavliotis, G. A. (2022). Noisy bounded confidence models for opinion dynamics: the effect of boundary conditions on phase transitions. IMA Journal of Applied Mathematics , 87(1):80--110

    work page 2022

  44. [44]

    and Podolskij, M

    Heidari, A. and Podolskij, M. (2025). Local asymptotic normality for discretely observed mckean-vlasov diffusions. arXiv preprint arXiv:2511.13366

    work page arXiv 2025

  45. [45]

    Hu, K., Ren, Z., Šiška, D., and Szpruch, . (2021). Mean-field Langevin dynamics and energy landscape of neural networks . Annales de l'Institut Henri Poincaré, Probabilités et Statistiques , 57(4):2043 -- 2065

    work page 2021

  46. [46]

    and Wang, F.-Y

    Huang, X. and Wang, F.-Y. (2019). Distribution dependent SDEs with singular coefficients. Stochastic Processes and their Applications , 129(11):4747--4770

    work page 2019

  47. [47]

    Iguchi, Y., Beskos, A., and Pavliotis, G. A. (2025). Parameter estimation for weakly interacting hypoelliptic diffusions. arXiv preprint arXiv:2508.04287

    work page arXiv 2025

  48. [48]

    Jasra, A., Maama, M., and Tempone, R. F. (2025). Parameter estimation for partially observed McKean--Vlasov diffusions. Royal Society Open Science , 12(12):251918

    work page 2025

  49. [49]

    and Wu, A

    Jasra, A. and Wu, A. (2025). Bayesian parameter estimation for partially observed McKean--Vlasov diffusions using multilevel Markov chain Monte Carlo . Statistics and Computing , 35(6):210

    work page 2025

  50. [50]

    Jourdain, B., M \'e l \'e ard, S., and Woyczynski, W. A. (2008). Nonlinear SDEs driven by L \'e vy processes and related PDEs . ALEA: Latin American Journal of Probability and Mathematical Statistics , 4:1--29

    work page 2008

  51. [51]

    Kasonga, R. A. (1990). Maximum likelihood theory for large interacting systems. SIAM Journal on Applied Mathematics , 50(3):865--875

    work page 1990

  52. [52]

    Kuramoto, Y. (1981). Rhythms and turbulence in populations of chemical oscillators. Physica A: Statistical Mechanics and its Applications , 106(1--2):128--143

    work page 1981

  53. [53]

    Lacker, D. (2018). On a strong form of propagation of chaos for McKean--Vlasov equations. Electronic Communications in Probability , 23:1--11

    work page 2018

  54. [54]

    Lacker, D. (2023). Hierarchies, entropy, and quantitative propagation of chaos for mean field diffusions. Probability and Mathematical Physics , 4(2):377--432

    work page 2023

  55. [55]

    and Le Flem, L

    Lacker, D. and Le Flem, L. (2023). Sharp uniform-in-time propagation of chaos. Probability Theory and Related Fields , 187(1--2):443--480

    work page 2023

  56. [56]

    and Lu, F

    Lang, Q. and Lu, F. (2023). Identifiability of interaction kernels in mean-field equations of interacting particles. Foundations of Data Science , 5(4):480--502

    work page 2023

  57. [57]

    Levanony, D., Shwartz, A., and Zeitouni, O. (1994). Recursive identification in continuous-time stochastic processes. Stochastic Processes and their Applications , 49(2):245--275

    work page 1994

  58. [58]

    and Wang, D

    Liu, Q. and Wang, D. (2016). Stein variational gradient descent: A general purpose Bayesian inference algorithm. In Proceedings of the 30th Annual Conference on Neural Information Processing Systems (NeurIPS 2016)

    work page 2016

  59. [59]

    Lu, F., Maggioni, M., and Tang, S. (2021). Learning interaction kernels in heterogeneous systems of agents from multiple trajectories. Journal of Machine Learning Research , 22(32):1--67

    work page 2021

  60. [60]

    Lu, F., Zhong, M., Tang, S., and Maggioni, M. (2019). Nonparametric inference of interaction laws in systems of agents from trajectory data. Proceedings of the National Academy of Sciences , 116(29):14424--14433

    work page 2019

  61. [61]

    and Poquet, C

    Lu c on, E. and Poquet, C. (2021). Periodicity induced by noise and interaction in the kinetic mean-field FitzHugh--Nagumo model. The Annals of Applied Probability , 31(2):561--593

    work page 2021

  62. [62]

    Malrieu, F. (2001). Logarithmic Sobolev inequalities for some nonlinear PDE 's. Stochastic Processes and their Applications , 95(1):109--132

    work page 2001

  63. [63]

    Malrieu, F. (2003). Convergence to equilibrium for granular media equations and their Euler schemes. The Annals of Applied Probability , 13(2):540--560

    work page 2003

  64. [64]

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

    work page 1966

  65. [65]

    Mei, S., Montanari, A., and Nguyen, P.-M. (2018). A mean field view of the landscape of two-layer neural networks. Proceedings of the National Academy of Sciences , 115(33):E7665--E7671

    work page 2018

  66. [66]

    M \'e l \'e ard, S. (1996). Asymptotic behaviour of some interacting particle systems; McKean--Vlasov and Boltzmann models. In Talay, D. and Tubaro, L., editors, Probabilistic Models for Nonlinear Partial Differential Equations , volume 1627 of Lecture Notes in Mathematics , pages 42--95. Springer, Berlin, Heidelberg

    work page 1996

  67. [67]

    Mishura, Y. S. and Veretennikov, A. Y. (2020). Existence and uniqueness theorems for solutions of McKean--Vlasov stochastic equations. Theory of Probability and Mathematical Statistics , 103:59--101

    work page 2020

  68. [68]

    A., and Ray, K

    Nickl, R., Pavliotis, G. A., and Ray, K. (2025). Bayesian nonparametric inference in McKean--Vlasov models. The Annals of Statistics , 53(1):170--193

    work page 2025

  69. [69]

    Oelschl \"a ger, K. (1984). A martingale approach to the law of large numbers for weakly interacting stochastic processes. The Annals of Probability , 12(2):458--479

    work page 1984

  70. [70]

    ksendal, B. (2003). Stochastic Differential Equations: An Introduction with Applications . Springer-Verlag, 6th edition

    work page 2003

  71. [71]

    A., Reich, S., and Zanoni, A

    Pavliotis, G. A., Reich, S., and Zanoni, A. (2025). Filtered data based estimators for stochastic processes driven by colored noise. Stochastic Processes and their Applications , 181:104558

    work page 2025

  72. [72]

    Pavliotis, G. A. and Zanoni, A. (2022). Eigenfunction martingale estimators for interacting particle systems and their mean field limit. SIAM Journal on Applied Dynamical Systems , 21(4):2338--2370

    work page 2022

  73. [73]

    Pavliotis, G. A. and Zanoni, A. (2024). A method of moments estimator for interacting particle systems and their mean field limit. SIAM/ASA Journal on Uncertainty Quantification , 12(2):262--288

    work page 2024

  74. [74]

    Pavliotis, G. A. and Zanoni, A. (2026). A Fourier-based inference method for learning interaction kernels in particle systems. SIAM Journal on Applied Mathematics , 86(2):615--643

    work page 2026

  75. [75]

    Rotskoff, G. M. and Vanden-Eijnden, E. (2022). Trainability and accuracy of artificial neural networks: An interacting particle system approach. Communications on Pure and Applied Mathematics , 75(9):1889--1935

    work page 2022

  76. [76]

    Sakaguchi, H., Shinomoto, S., and Kuramoto, Y. (1988). Phase transitions and their bifurcation analysis in a large population of active rotators with mean-field coupling. Progress of Theoretical Physics , 79(3):600--607

    work page 1988

  77. [77]

    Sharrock, L. (2022a). On the Theory and Applications of Stochastic Gradient Descent in Continuous Time . PhD thesis, Imperial College London

    work page

  78. [78]

    Sharrock, L. (2022b). Two-timescale stochastic approximation for bilevel optimisation problems in continuous-time models. In Proceedings of the 39th International Conference on Machine Learning (ICML 2022): Workshop on Continuous Time Methods for Machine Learning

    work page 2022

  79. [79]

    and Kantas, N

    Sharrock, L. and Kantas, N. (2022). Joint online parameter estimation and optimal sensor placement for the partially observed stochastic advection diffusion equation. SIAM/ASA Journal on Uncertainty Quantification , 10(1):55--95

    work page 2022

  80. [80]

    and Kantas, N

    Sharrock, L. and Kantas, N. (2023). Two-timescale stochastic gradient descent in continuous time with applications to joint online parameter estimation and optimal sensor placement. Bernoulli , 29(2):1137--1165

    work page 2023

Showing first 80 references.