On the exit-problem for self-interacting diffusions

We study the exit-time from a domain of a self-interacting diffusion, where the Brownian motion is replaced by $σB_t$ for a constant $σ$. The first part of this work consists in showing that the rate of convergence (of the occupation measure of the self-interacting process toward some explicit Gibbs measure) previously obtained in \cite{kk-ejp} for a convex confinment potential $V$ and a convex interaction potential can be bounded uniformly with respect to $σ$. Then, we prove an Arrhenius-type law for the first exit-time from a domain (satisfying classical hypotheses of Freidlin-Wentzell theory).

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