Long term behaviour of self-interacting diffusions

 In this talk, I will present the results I obtained in my Ph.d-thesis about the long term behaviour of strong self-interacting diffusions on compact manifolds (the unit circle S^1 is always considered). Roughly speaking, strong self-interacting diffusions (as considered in this talk) are time continuous stochastic processes which solve a homogeneous stochastic differential equation whose drift part is evolving in time according to the whole past history of the process. If it is in such a way that it tends to push the diffusing particle away from the most visited sites, the system is said self-repelling; and if it tends to go back to the most visited sites, the system is said self-attracting. Some results to be presented in this talk are joint work with Michel Benaim, Ioana Ciotir and Pierre Monmarché.