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Nonlinear self-stabilizing processes - II: Convergence to invariant probability

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  • Benachour, S.
  • Roynette, B.
  • Vallois, P.

Abstract

We now analyze the asymptotic behaviour of Xt, as t approaches infinity, X being solution of where [beta] is a given odd and increasing Lipschitz-continuous function with polynomial growth. We prove with additional assumptions on [beta] that Xt converges in distribution to the invariant probability measure associated with Eq. 1.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:75:y:1998:i:2:p:203-224
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    1. 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.
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    Cited by:

    1. Malrieu, F., 2001. "Logarithmic Sobolev inequalities for some nonlinear PDE's," Stochastic Processes and their Applications, Elsevier, vol. 95(1), pages 109-132, September.
    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. 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.
    4. 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.
    5. 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.

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