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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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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DOI: https://doi.org/10.1007/s004400100161
Keywords
- Differential Equation
- Manifold
- Probability Measure
- Asymptotic Behavior
- Riemannian Manifold