Exit-problem for a class of non-Markov processes with path dependency

Abstract

We study the exit-time of a self-interacting diffusion from an open domain $G\subset {\mathbb{R}}^d$. In particular, we consider the equation $d{X}_t=-\left(\nabla V(X_t)+\frac{1}{t}{\int }_0^t\nabla F(X_t-X_s)ds\right)dt+\ddot{I}ƒd{W}_t.$ We are interested in the small-noise ($\ddot{I}ƒ\to 0$) behaviour of the exit-time from the potentials' domain of attraction. In this work rather weak assumptions on the potentials $V$ and $F$, and on the domain $G$ are considered. In particular, we do not assume $V$ nor $F$ to be either convex or concave, which covers a wide range of self-attracting and self-repelling stochastic processes possibly moving in a complex multi-well landscape. The Large Deviation Principle for the Self-interacting diffusion with generalized initial conditions is established. The main result of the paper states that, under some assumptions on the potentials $V$ and $F$, and on the domain $G$, the Kramers' type law for the exit-time holds. Finally, we provide a result concerning the exit-location of the diffusion.