Establishing the LDP for the SID: What Does It Mean?

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 2.1 Establishing the LDP for the SID In this section, we prove LDP for Self-Interacting diffusions of the type (1.7). First, we introduce the following group of main assumptions. Assumptions A-1. Consider the following theorem that is the main result of the section. Use Lipschitz continuity of ∇V , ∇F (Assumption A-1.ii) and get: Therefore, by Grönwall’s inequality it means the continuity of the map G for any possible x ∈ X. 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 2.1 Establishing the LDP for the SID In this section, we prove LDP for Self-Interacting diffusions of the type (1.7). First, we introduce the following group of main assumptions. Assumptions A-1. Assumptions A-1. Consider the following theorem that is the main result of the section. Use Lipschitz continuity of ∇V , ∇F (Assumption A-1.ii) and get: Therefore, by Grönwall’s inequality it means the continuity of the map G for any possible x ∈ X. 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.