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Non-uniqueness of stationary measures for self-stabilizing processes

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  • Herrmann, S.
  • Tugaut, J.

Abstract

We investigate the existence of invariant measures for self-stabilizing diffusions. These stochastic processes represent roughly the behavior of some Brownian particle moving in a double-well landscape and attracted by its own law. This specific self-interaction leads to nonlinear stochastic differential equations and permits pointing out singular phenomena like non-uniqueness of associated stationary measures. The existence of several invariant measures is essentially based on the non-convex environment and requires generalized Laplace's method approximations.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:120:y:2010:i:7:p:1215-1246
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    References listed on IDEAS

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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. 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. 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.
    2. Bashiri, K. & Menz, G., 2021. "Metastability in a continuous mean-field model at low temperature and strong interaction," Stochastic Processes and their Applications, Elsevier, vol. 134(C), pages 132-173.
    3. 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.

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