A Generalization of Our Exit-problem for a Class of Non-Markov Processes With Path Dependency Paper

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

4 Generalization

In this section we present the possible generalisation of Assumptions A-1 and A-2 as well as the exit-time result for this more general process. In this paper we did not talk in details about the problem of existence and uniqueness of the self-interacting diffusion. While in the Lipschitz case (Assumption A-1) these questions are standard, locally Lipschitzness of the potentials add some complication. In [AdMKT23], the authors provide a proof for existence and uniqueness of self-interacting diffusion under the following assumptions that we take as a baseline:

Assumptions A-1′.

It is worth noting that Assumption A-1′ .ii is a combination of locally Lipschitz and polynomial growth conditions. Assumption A-1′ .iii has been introduced in [AdMKT23] to control the Lyapunov functional of the diffusion process to ensure that there is no explosion within a finite time. It is important to mention that, if required, this assumption can be replaced by another one that ensures the process’s global existence and uniqueness without any impact on the exit-time result.

We can also relax Assumptions A-2. Indeed, the domain G does not necessarily have to be bounded, but it is necessary to add another assumption on the level sets instead. Consider:

Assumptions A-2′.

Under these assumptions we can obtain Kramers’ type law and the exit location result.

Acknowledgments: This work has been supported by the French ANR grant METANOLIN (ANR-19-CE40-0009).

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