Jan 21, 1970
Exit-Problem for a Class of Non-Markov Processes With Path Dependency: Proof of the Main Theorem by@classpath
Exit-Problem for a Class of Non-Markov Processes With Path Dependency: Proof of the Main Theorem
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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.Company Mentioned
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1.2 Some remarks on dynamics and initial condition
2.1 Establishing the LDP for the SID
2.2 Results related to the LDP
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.
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