On the exit-problem for self-interacting diffusions

Ashot Aleksian (1) , Pierre del Moral (2) , Aline Kurtzmann (3) , Julian Tugaut (1)

1 PSPM - Probabilités, statistique, physique mathématique
2 ASTRAL - Méthodes avancées d’apprentissage statistique et de contrôle
3 IECL - Institut Élie Cartan de Lorraine

We study the exit-time from a domain of a self-interacting diffusion, where the Brownian motion is replaced by $\sigma B_t$ for a constant $\sigma$. 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 $\sigma$. Then, we prove an Arrhenius-type law for the first exit-time from a domain (satisfying classical hypotheses of Freidlin-Wentzell theory).

Mots clés: Kramers' law; Deterministic flow; Exit time; Self-interacting diffusion

Domaines: Probabilités [math.PR]

Ashot Aleksian, Pierre del Moral, Aline Kurtzmann, Julian Tugaut. On the exit-problem for self-interacting diffusions. 2022. ⟨hal-01901145v2⟩