Electronic Journal of Probability

Asymptotically exponential hitting times and metastability: a pathwise approach without reversibility

Roberto Fernandez, Francesco Manzo, Francesca Nardi, and Elisabetta Scoppola

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Abstract

We study the hitting times of Markov processes to target set $G$, starting from a reference configuration $x_0$ or its basin of attraction. The configuration $x_0$ can correspond to the bottom of a (meta)stable well, while the target $G$ could be either a set of saddle (exit) points of the well, or a set of further (meta)stable configurations. Three types of results are reported: (1) A general theory is developed, based on the path-wise approach to metastability, which has three important attributes. First, it is general in that it does not assume reversibility of the process, does not focus only on hitting times to rare events and does not assume a particular starting measure. Second, it relies only on the natural hypothesis that the mean hitting time to $G$ is asymptotically longer than the mean recurrence time to $x_0$ or $G$. Third, despite its mathematical simplicity, the approach yields precise and explicit bounds on the corrections to exponentiality. (2) We compare and relate different metastability conditions proposed in the literature so to eliminate potential sources of confusion. This is specially relevant for evolutions of infinite-volume systems, whose treatment depends on whether and how relevant parameters (temperature, fields) are adjusted. (3) We introduce the notion of early asymptotic exponential behavior to control time scales asymptotically smaller than the mean-time scale. This control is particularly relevant for systems with unbounded state space where nucleations leading to exit from metastability can happen anywhere in the volume. We provide natural sufficient conditions on recurrence times for this early exponentiality to hold and show that it leads to estimations of probability density functions.

Article information

Source
Electron. J. Probab., Volume 20 (2015), paper no. 122, 37 pp.

Dates
Received: 10 July 2014
Accepted: 11 October 2015
First available in Project Euclid: 4 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1465067228

Digital Object Identifier
doi:10.1214/EJP.v20-3656

Mathematical Reviews number (MathSciNet)
MR3425542

Zentralblatt MATH identifier
1329.60269

Subjects
Primary: 60J27-60J28-82C05

Keywords
Asymptotic exponential behavior Hitting times Metastability Tunneling

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Fernandez, Roberto; Manzo, Francesco; Nardi, Francesca; Scoppola, Elisabetta. Asymptotically exponential hitting times and metastability: a pathwise approach without reversibility. Electron. J. Probab. 20 (2015), paper no. 122, 37 pp. doi:10.1214/EJP.v20-3656. https://projecteuclid.org/euclid.ejp/1465067228


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