Exit-Problem for a Class of Non-Markov Processes With Path Dependency

Table of Links

Abstract and 1 Introduction

1.1 State of the art

1.2 Some remarks on dynamics and initial condition

1.3 Outline of the paper

1.4 List of notations

2 Large Deviation Principle

2.1 Establishing the LDP for the SID

2.2 Results related to the LDP

2.3 Compactness results

3 Exit-time

3.1 Auxiliary results

3.2 Proof of the main theorem

3.3 Proofs of auxiliary lemmas

4 Generalization and References

Abstract

We are interested in small-noise ( σ → 0) behaviour of exit-time from the potentials’ domain of attraction. In this work rather weak assumptions on potentials V and F, and on 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-avoiding stochastic processes possibly moving in a complex multi-well landscape. The Large Deviation Principle for Self-interacting diffusion with generalized initial conditions is established. The main result of the paper states that, under some assumptions on potentials V and F, and on domain G, Kramers’ type law for the exit-time holds. Finally, we provide a result concerning the exit-location of the diffusion.

1 Introduction