2022
DOI: 10.48550/arxiv.2207.02588
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Metastable $Γ$-expansion of finite state Markov chains level two large deviations rate functions

Abstract: We examine two analytical characterisation of the metastable behavior of a Markov chain. The first one expressed in terms of its transition probabilities, and the second one in terms of its large deviations rate functional.Consider a sequence of continuous-time Markov chains (Xnp , 1 ≤ p ≤ q.

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Cited by 5 publications
(13 citation statements)
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References 49 publications
(77 reference statements)
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“…On each time scale, the irreducible components merge with each other and become a single collection on the next time scale. (4) Continuing the previous remark, this hierarchical structure of metastability is closely related to the so-called Γ-expansion approach to metastability [6,30,34]. Briefly, the idea is as follows.…”
Section: 4mentioning
confidence: 84%
See 1 more Smart Citation
“…On each time scale, the irreducible components merge with each other and become a single collection on the next time scale. (4) Continuing the previous remark, this hierarchical structure of metastability is closely related to the so-called Γ-expansion approach to metastability [6,30,34]. Briefly, the idea is as follows.…”
Section: 4mentioning
confidence: 84%
“…The hierarchy of metastable time scales is closely related to the so-called Γ-expansion approach to metastability [6,30,34], a topic recently well recognized in the community. Ongoing work tries to characterize this Γ-expansion of level-two large deviation rate functionals subject to the reversible inclusion process; refer to Remark 2.12-(4) for more details.…”
mentioning
confidence: 99%
“…We stress that (i) implies a quantitative scale separation between the N 0 slow modes, corresponding to the metastable tunneling times, and all the other modes, corresponding to fast relaxations to local equilibria. In principle it is also possible to refine the analysis of the fast modes revealing the full hierarchy of scales governing the dynamics in the small ε regime, see [19] for a -convergence formulation in continuous space setting and the recent [4].…”
Section: Results On the Diffusion Operator L εmentioning
confidence: 99%
“…Main assumption. To examine the convergence of θ n I n for some sequence θ n → ∞, we introduce a natural hypothesis on the jump rates proposed in [4] and adopted in [5,11,23].…”
Section: Notation and Resultsmentioning
confidence: 99%
“…The proof o Theorem 2.3 relies on [4,23] where the metastable behavior of the sequence X (n) t has been investigated. The expansion (1.6) has been derived for reversible diffusions in [12] and for reversible finite state Msrkov chains in [5]. It should be a universal property of Markov chains and should hold for dynamics whose state space depend on n and which exhibit a metastable behavior at different time-scales.…”
Section: Introductionmentioning
confidence: 99%