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Self-stabilizing processes in multi-wells landscape in Rd-convergence

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  • Tugaut, Julian

Abstract

Self-stabilizing processes are inhomogeneous diffusions in which the law of the process intervenes in the drift. If the external force is the gradient of a convex potential, it has been proved that the process converges towards the unique invariant probability as the time goes to infinity. However, in a previous article, we established that the diffusion may admit several invariant probabilities, provided that the external force derives from a non-convex potential. We here provide results about the limiting values of the family {μt;t≥0}, μt being the law of the diffusion. Moreover, we establish the weak convergence under an additional hypothesis.

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  • 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.
    • Handle: RePEc:eee:spapps:v:123:y:2013:i:5:p:1780-1801
    DOI: 10.1016/j.spa.2012.12.003
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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. Graham, Carl, 1992. "McKean-Vlasov Ito-Skorohod equations, and nonlinear diffusions with discrete jump sets," Stochastic Processes and their Applications, Elsevier, vol. 40(1), pages 69-82, February.
    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. 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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