Exit-Problem for a Class of Non-Markov Processes With Path Dependency: State of the Art

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

1.1 State of the art

Similar systems with path-interaction behaviour have already been studied by numerous researchers for more than 30 years. In most of the papers the long-time behaviour of the process is considered. One of the first mathematical descriptions of such a process under the name of Self-avoiding random walk is presented by J.R. Norris, L.C.G. Rogers, and D. Williams in [NRW87]. Some stochastic properties of the process along with some long-time behaviour results in a particular setting were shown. The main difference between the system considered there and ours is that in [NRW87] there is no renormalization of the interaction term with time. In [DR92] R.T. Durrett and L.C.G. Rogers introduce a similar to the previous paper system that aims to model the shape of a growing polymer:

where F is regular and V is convex at infinity. In this paper the ergodic properties of X were studied as well as certain conditions on V and F that guarantee almost sure convergence of µt. This work generalizes the previous ones of Benaïm and co-authors.

In [Tug12] and [Tug16], J. Tugaut studies the exit-time problem for SSD in convex landscape with convex interaction as it was presented in [HIP08]. In these papers he establishes the Kramers’ type law avoiding using large deviations principle. Instead, the author uses various analytical and coupling methods to deal with the problem, thus simplifying calculations. This work was continued in [Tug18] and [Tug19] where the same techniques were used to establish the Kramers’ type law in the case of confinement potential V that is of the double-well form. However, exit-time for the SSD with general assumptions (as in Freidlin-Wentzell theory for Itô diffusion) is still an open problem.

Exit-time problem was also studied for the case of self-interacting diffusion. In [AdMKT23], A. Aleksian, P. Del Moral, A. Kurtzmann, and J. Tugaut prove Kramers’ type law for SID in which both interaction and confinement potentials V and F are convex. This nice property of potentials was used by the authors in order to prove this exit-time result by applying analytical and coupling techniques similar to the ones used in [Tug16]. Since the goal of the current paper is to establish some exit-time result for SID with more general, than in [AdMKT23], assumptions, we have to use a different approach. In our case, that consists in proving the Large Deviation Principle and restoring the logic of Freidlin-Wentzell theory for SID.