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Nonlinear self-stabilizing processes - I Existence, invariant probability, propagation of chaos

Author

Listed:
  • Benachour, S.
  • Roynette, B.
  • Talay, D.
  • Vallois, P.

Abstract

Taking an odd, non-decreasing function [beta], we consider the (nonlinear) stochastic differential equation and we prove the existence and uniqueness of solution of Eq. E , where and (Bt; t[greater-or-equal, slanted]0) is a one-dimensional Brownian motion, B0=0. We show that Eq. E admits a stationary probability measure and investigate the link between Eq. E and the associated system of particles.

Suggested Citation

  • Benachour, S. & Roynette, B. & Talay, D. & Vallois, P., 1998. "Nonlinear self-stabilizing processes - I Existence, invariant probability, propagation of chaos," Stochastic Processes and their Applications, Elsevier, vol. 75(2), pages 173-201, July.
    • Handle: RePEc:eee:spapps:v:75:y:1998:i:2:p:173-201
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    1. Benachour, S. & Roynette, B. & Vallois, P., 1998. "Nonlinear self-stabilizing processes - II: Convergence to invariant probability," Stochastic Processes and their Applications, Elsevier, vol. 75(2), pages 203-224, July.
    2. Adams, Daniel & dos Reis, Gonçalo & Ravaille, Romain & Salkeld, William & Tugaut, Julian, 2022. "Large Deviations and Exit-times for reflected McKean–Vlasov equations with self-stabilising terms and superlinear drifts," Stochastic Processes and their Applications, Elsevier, vol. 146(C), pages 264-310.
    3. Malrieu, F., 2001. "Logarithmic Sobolev inequalities for some nonlinear PDE's," Stochastic Processes and their Applications, Elsevier, vol. 95(1), pages 109-132, September.
    4. Liu, Huoxia & Lin, Judy Yangjun, 2023. "Stochastic McKean–Vlasov equations with Lévy noise: Existence, attractiveness and stability," Chaos, Solitons & Fractals, Elsevier, vol. 177(C).
    5. 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.
    6. Yulin Song, 2020. "Gradient Estimates and Exponential Ergodicity for Mean-Field SDEs with Jumps," Journal of Theoretical Probability, Springer, vol. 33(1), pages 201-238, March.
    7. Sharrock, Louis & Kantas, Nikolas & Parpas, Panos & Pavliotis, Grigorios A., 2023. "Online parameter estimation for the McKean–Vlasov stochastic differential equation," Stochastic Processes and their Applications, Elsevier, vol. 162(C), pages 481-546.
    8. Tugaut, Julian, 2013. "Self-stabilizing processes in multi-wells landscape in Rd-convergence," Stochastic Processes and their Applications, Elsevier, vol. 123(5), pages 1780-1801.
    9. Herrmann, S. & Tugaut, J., 2010. "Non-uniqueness of stationary measures for self-stabilizing processes," Stochastic Processes and their Applications, Elsevier, vol. 120(7), pages 1215-1246, July.
    10. Julian Tugaut, 2014. "Self-stabilizing Processes in Multi-wells Landscape in ℝ d -Invariant Probabilities," Journal of Theoretical Probability, Springer, vol. 27(1), pages 57-79, March.

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