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

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.


This content originally appeared on HackerNoon and was authored by Class Path

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

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:::info This paper is available on arxiv under CC BY-SA 4.0 DEED license.

:::

:::info Authors:

(1) Ashot Aleksian, Université Jean Monnet, Institut Camille Jordan, 23, rue du docteur Paul Michelon, CS 82301, 42023 Saint-Étienne Cedex 2, France;

(2) Aline Kurtzmann, Université de Lorraine, CNRS, Institut Elie Cartan de Lorraine UMR 7502, Vandoeuvre-lès-Nancy, F-54506, France;

(3) Julian Tugaut, Université Jean Monnet, Institut Camille Jordan, 23, rue du docteur Paul Michelon, CS 82301, 42023 Saint-Étienne Cedex 2, France.

:::

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This content originally appeared on HackerNoon and was authored by Class Path


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