Self-interacting diffusions

Abstract

This paper is concerned with a general class of self-interacting diffusions {Xt}t≥0 living on a compact Riemannian manifold M. These are solutions to stochastic differential equations of the form: dXt = Brownian increments + drift term depending on Xt and μt, the normalized occupation measure of the process. It is proved that the asymptotic behavior of {μt} can be precisely related to the asymptotic behavior of a deterministic dynamical semi-flow Φ = {Φt}t≥0 defined on the space of the Borel probability measures on M. In particular, the limit sets of {μt} 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 {μt} 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.