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A Hilbertian approach for fluctuations on the McKean-Vlasov model

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  • Fernandez, Begoña
  • Méléard, Sylvie

Abstract

We consider the sequence of fluctuation processes associated with the empirical measures of the interacting particle system approximating the d-dimensional McKean-Vlasov equation and prove that they are tight as continuous processes with values in a precise weighted Sobolev space. More precisely, we prove that these fluctuations belong uniformly (with respect to the size of the system and to time) to W-(1+D), 2D0 and converge in C([0, T], W-(2+2D), D0) to a Ornstein-Uhlenbeck process obtained as the solution of a Langevin equation in W-(4+2D), D0, where D is equal to 1 + [d/2]. It appears in the proofs that the spaces W-(1 --> D), 2D0 and W-(2-2D), D0 are minimal Sobolev spaces in which to immerse the fluctuations, which was our aim following a physical point of view.

Suggested Citation

  • Fernandez, Begoña & Méléard, Sylvie, 1997. "A Hilbertian approach for fluctuations on the McKean-Vlasov model," Stochastic Processes and their Applications, Elsevier, vol. 71(1), pages 33-53, October.
    • Handle: RePEc:eee:spapps:v:71:y:1997:i:1:p:33-53
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    References listed on IDEAS

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    1. Hitsuda, Masuyuki & Mitoma, Itaru, 1986. "Tightness problem and stochastic evolution equation arising from fluctuation phenomena for interacting diffusions," Journal of Multivariate Analysis, Elsevier, vol. 19(2), pages 311-328, August.
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    Cited by:

    1. Amorino, Chiara & Heidari, Akram & Pilipauskaitė, Vytautė & Podolskij, Mark, 2023. "Parameter estimation of discretely observed interacting particle systems," Stochastic Processes and their Applications, Elsevier, vol. 163(C), pages 350-386.
    2. Genon-Catalot, Valentine & Larédo, Catherine, 2021. "Probabilistic properties and parametric inference of small variance nonlinear self-stabilizing stochastic differential equations," Stochastic Processes and their Applications, Elsevier, vol. 142(C), pages 513-548.
    3. Heldman, M. & Isaacson, S.A. & Ma, J. & Spiliopoulos, K., 2024. "Fluctuation analysis for particle-based stochastic reaction–diffusion models," Stochastic Processes and their Applications, Elsevier, vol. 167(C).
    4. Sirignano, Justin & Spiliopoulos, Konstantinos, 2020. "Mean field analysis of neural networks: A central limit theorem," Stochastic Processes and their Applications, Elsevier, vol. 130(3), pages 1820-1852.

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