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 2.2 Results related to the LDP The following lemma generalizes the large deviation principle for the case of converging initial conditions. and that proves the first inequality. One can prove the second inequality the same way. As was pointed out before, convergence of measures in Wasserstein distance gives convergence of respective integrals, since ∇F is Lipschitz continuous [Vil09, Theorem 6.9]. As was pointed out before, lower semicontinuity guarantees that infima of a function are achieved over compact sets. We summarise this property by the following corollary. This paper is available on arxiv under CC BY-SA 4.0 DEED license. 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. Table of Links Abstract and 1 Introduction Abstract and 1 Introduction 1.1 State of the art 1.1 State of the art 1.2 Some remarks on dynamics and initial condition 1.2 Some remarks on dynamics and initial condition 1.3 Outline of the paper 1.3 Outline of the paper 1.4 List of notations 1.4 List of notations 2 Large Deviation Principle 2 Large Deviation Principle 2.1 Establishing the LDP for the SID 2.1 Establishing the LDP for the SID 2.2 Results related to the LDP 2.2 Results related to the LDP 2.3 Compactness results 2.3 Compactness results 3 Exit-time 3 Exit-time 3.1 Auxiliary results 3.1 Auxiliary results 3.2 Proof of the main theorem 3.2 Proof of the main theorem 3.3 Proofs of auxiliary lemmas 3.3 Proofs of auxiliary lemmas 4 Generalization and References 4 Generalization and References 2.2 Results related to the LDP The following lemma generalizes the large deviation principle for the case of converging initial conditions. and that proves the first inequality. One can prove the second inequality the same way. As was pointed out before, convergence of measures in Wasserstein distance gives convergence of respective integrals, since ∇F is Lipschitz continuous [Vil09, Theorem 6.9]. As was pointed out before, lower semicontinuity guarantees that infima of a function are achieved over compact sets. We summarise this property by the following corollary. This paper is available on arxiv under CC BY-SA 4.0 DEED license. This paper is available on arxiv under CC BY-SA 4.0 DEED license. available on arxiv 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. Authors: 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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