The ODE method for some self-interacting diffusions on non-compact spaces

Self-interacting diffusions are solutions to SDEs with a drift term depending on the process and its normalized occupation measure ${\mu }_t$ (via an interaction potential and a confinement potential). We establish a relation between the asymptotic behavior of ${\mu }_t$ and the asymptotic behavior of a deterministic dynamical flow (defined on the space of the Borel probability measures). We extend previous results on ${\mathbb{R}}^d$ or more generally a smooth complete connected Riemannian manifold without boundary. We will also give some sufficient conditions for the convergence of ${\mu }_t$. Finally, we will illustrate our study with an example on ${\mathbb{R}}^2$.