Self-interacting diffusions

This paper is concerned with a general class of self-interacting diffusions [X-t}(tgreater than or equal to0) living oil 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 mu(t), the normalized occupation measure of the process. It is proved that the asymptotic behavior of {mu(t)} can be precisely related to the asymptotic behavior of a deterministic dynamical semi-flow Phi = {Phi(t)}(tgreater than or equal to0) defined on the space of the Borel probability measures on M. In particular, the limit sets of {mu(t)} are proved to be almost surely attractor free sets for Phi. 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 {mu(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.