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 3.2 Proof of the main theorem 3.2.1 Kramers’ law Given the results of Lemmas 3.2 and 3.5, we expect our process to spend most of its time near a with σ small enough. In order to have more information about this behaviour, we introduce the following stopping times That gives us: Therefore, by induction in k, we get: By Markov’s inequality, we get: We remark that by Lemmas 3.1, 3.5 and derivations above, we have This concludes the proof. 3.2.2 Exit-location 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 3.2 Proof of the main theorem 3.2.1 Kramers’ law Given the results of Lemmas 3.2 and 3.5, we expect our process to spend most of its time near a with σ small enough. In order to have more information about this behaviour, we introduce the following stopping times That gives us: Therefore, by induction in k, we get: By Markov’s inequality, we get: We remark that by Lemmas 3.1, 3.5 and derivations above, we have This concludes the proof. 3.2.2 Exit-location 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.

Part of HackerNoon's growing list of open-source research papers, promoting free access to academic material.

Abstract

Exit-Problem for a Class of Non-Markov Processes With Path Dependency: Proof of the Main Theorem

About Author

Comments

TOPICS

THIS ARTICLE WAS FEATURED IN

Related Stories

What We Mean By the Large Deviation Principle (LDP)

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

Results Related to the LDP: An Explanation

The Compactness Results of Our Non-Markov Processes With Path Dependency Paper

Exit-Problem for a Class of Non-Markov Processes With Path Dependency: A Peek at the Exit-Time

Exit-Problem for a Class of Non-Markov Processes With Path Dependency: The Auxiliary Results

What We Mean By the Large Deviation Principle (LDP)

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

Results Related to the LDP: An Explanation

The Compactness Results of Our Non-Markov Processes With Path Dependency Paper

Exit-Problem for a Class of Non-Markov Processes With Path Dependency: A Peek at the Exit-Time

Exit-Problem for a Class of Non-Markov Processes With Path Dependency: The Auxiliary Results

Light-Mode

Classic

Newspaper

Dark-Mode

Neon Noir

Minty

HN StartUps