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Exact Results on the First Hitting via Conditional Strong Quasi-Stationary Times and Applications to Metastability

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Abstract

In the setting of non-reversible Markov chains on finite or countable state space, exact results on the distribution of the first hitting time to a given set G are obtained. A new notion of “conditional strong quasi stationary time” is introduced to describe the local relaxation time. This time is defined via a generalization of the strong stationary time. Rarity of the target set G is not required and the initial distribution can be completely general. The results clarify the the role played by the initial distribution on the exponential law; they are used to give a general notion of metastability and to discuss the relation between the exponential distribution of the first hitting time and metastability.

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Acknowledgements

We thank Amine Asselah, Nils Berglund, Pietro Caputo, Frank den Hollander, Roberto Fernandez and Alexandre Gaudillière for many fruitful discussions. This work was partially supported by the A*MIDEX project (n. ANR-11-IDEX-0001-02) funded by the “Investissements d’Avenir” French Government program, managed by the French National Research Agency (ANR). E.S. has been supported by the PRIN 20155PAWZB Large Scale Random Structures.

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  1. Dipartimento di Matematica e Fisica, University of Roma “Roma Tre” Largo San Murialdo, 1 - 00146, Roma, Italy

    F. Manzo & E. Scoppola

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  1. F. Manzo
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  2. E. Scoppola
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Correspondence to E. Scoppola.

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Manzo, F., Scoppola, E. Exact Results on the First Hitting via Conditional Strong Quasi-Stationary Times and Applications to Metastability. J Stat Phys 174, 1239–1262 (2019). https://doi.org/10.1007/s10955-019-02233-3

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  • DOI: https://doi.org/10.1007/s10955-019-02233-3

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