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Self-interacting diffusions
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  • Published: January 2002

Self-interacting diffusions

  • Michel Benaïm1,
  • Michel Ledoux2 &
  • Olivier Raimond3 

Probability Theory and Related Fields volume 122, pages 1–41 (2002)Cite this article

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  • 44 Citations

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Abstract.

 This paper is concerned with a general class of self-interacting diffusions {X t } t ≥0 living on a compact Riemannian manifold M. These are solutions to stochastic differential equations of the form : dX t = Brownian increments + drift term depending on X t and μ t , the normalized occupation measure of the process. It is proved that the asymptotic behavior of {μ t } can be precisely related to the asymptotic behavior of a deterministic dynamical semi-flow Φ = {Φ t } t ≥0 defined on the space of the Borel probability measures on M. In particular, the limit sets of {μ t } are proved to be almost surely attractor free sets for Φ. These results are applied to several examples of self-attracting/repelling diffusions on the n-sphere. For instance, in the case of self-attracting diffusions, our results apply to prove that {μ t } can either converge toward the normalized Riemannian measure, or to a gaussian measure, depending on the value of a parameter measuring the strength of the attraction.

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Authors and Affiliations

  1. Département de Mathématiques, UFR des Sciences et Techniques, Université Cergy-Pontoise, 2 av. Adolphe Chauvin, 95302 Cergy-Pontoise Cedex, France. e-mail: Michel.Benaim@math.u-cergy.fr, , , , , , FR

    Michel Benaïm

  2. Laboratoire de Statistique et Probabilités, Université Paul Sabatier, Toulouse, France, , , , , , FR

    Michel Ledoux

  3. Laboratoire de Modélisation Stochastique et Statistique, Université Paris Sud, France, , , , , , FR

    Olivier Raimond

Authors
  1. Michel Benaïm
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  2. Michel Ledoux
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  3. Olivier Raimond
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Additional information

Received: 21 July 2000 / Revised version: 12 December 2000 / Published online: 15 October 2001

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Benaïm, M., Ledoux, M. & Raimond, O. Self-interacting diffusions. Probab Theory Relat Fields 122, 1–41 (2002). https://doi.org/10.1007/s004400100161

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  • Issue Date: January 2002

  • DOI: https://doi.org/10.1007/s004400100161

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Keywords

  • Differential Equation
  • Manifold
  • Probability Measure
  • Asymptotic Behavior
  • Riemannian Manifold
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