What We Mean By the Large Deviation Principle (LDP)

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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

1.3 Outline of the paper

Section 2 is dedicated to the Large deviation principle. First, we define what we mean by the large deviation principle (LDP), then we establish, under the set of assumptions A-1, LDP for self-interacting diffusion in Theorem 2.2. Next, some useful (for exit-time problem) results related to LDP are presented.

Section 3 deals with the exit-time problem. First, we present the set of assumptions A-2 that contains assumptions on domain G, its boundary and stability properties. We discuss the role that each of these assumptions play in later proof and present some examples of confinement and interaction potentials along with possible domains G that one can choose. Then, we present the main result of the paper that is Theorem 1.1. This result is followed by Section 3.1 where we present auxiliary lemmas that are later used in Section 3.2 to prove the main Theorem 1.1. These lemmas are later proved in Section 3.3.

Section 4 is focused on the generalization of Assumptions A-1 and A-2. In this section, we examine the scenario of a locally Lipschitz continuous drift term (∇V and ∇F) and an unbounded domain G from which we want to exit.