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Self-stabilizing Processes in Multi-wells Landscape in ℝ d -Invariant Probabilities

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  • Julian Tugaut

    (Universität Bielefeld)

Abstract

The aim of this work is to analyze the stationary measures for a particular class of non-Markovian diffusions, the self-stabilizing processes. All the trajectories of such a process attract each other. This permits to exhibit a non-uniqueness of the stationary measures in the one-dimensional case, see Herrmann and Tugaut (Stoch. Process. Their Appl. 120(7):1215–1246, 2010). In this paper, the extension to general multi-wells lansdcape in general dimension is provided. Moreover, the approach for investigating this problem is different and needs fewer assumptions. The small-noise limit behavior of the invariant probabilities is also given.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:jotpro:v:27:y:2014:i:1:d:10.1007_s10959-012-0435-2
    DOI: 10.1007/s10959-012-0435-2
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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. 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.
    3. 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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