## The Annals of Applied Probability

### Conditioned, quasi-stationary, restricted measures and escape from metastable states

#### Abstract

We study the asymptotic hitting time $\tau^{(n)}$ of a family of Markov processes $X^{(n)}$ to a target set $G^{(n)}$ when the process starts from a “trap” defined by very general properties. We give an explicit description of the law of $X^{(n)}$ conditioned to stay within the trap, and from this we deduce the exponential distribution of $\tau^{(n)}$. Our approach is very broad—it does not require reversibility, the target $G$ does not need to be a rare event and the traps and the limit on $n$ can be of very general nature—and leads to explicit bounds on the deviations of $\tau^{(n)}$ from exponentially. We provide two nontrivial examples to which our techniques directly apply.

#### Article information

Source
Ann. Appl. Probab., Volume 26, Number 2 (2016), 760-793.

Dates
Revised: February 2015
First available in Project Euclid: 22 March 2016

https://projecteuclid.org/euclid.aoap/1458651819

Digital Object Identifier
doi:10.1214/15-AAP1102

Mathematical Reviews number (MathSciNet)
MR3476624

Zentralblatt MATH identifier
1339.60110

#### Citation

Fernandez, R.; Manzo, F.; Nardi, F. R.; Scoppola, E.; Sohier, J. Conditioned, quasi-stationary, restricted measures and escape from metastable states. Ann. Appl. Probab. 26 (2016), no. 2, 760--793. doi:10.1214/15-AAP1102. https://projecteuclid.org/euclid.aoap/1458651819

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