On metastability
The goal of this work is to introduce a quantity which measures the metastability of an absorbed Markov chain on a finite set, without referring to the asymptotic behaviour of some parameter such as a temperature. This quantity is presented as a spectral criterion, in the sense that it is defined from the spectrum of sub-Markovian chains which are obtained from the initial one by adding a regeneration mechanism according to the quasi-stationary state and then by removing one point in the boundary of the chain (i.e., the points at which the absorbing rate is positive). The metastability is quantified in terms of the difference between the law of the absorbing time and an exponential law. The first main result is that the distance between the repartition functions of these two laws is bounded uniformly in time and in the initial condition by the spectral quantity times the absorbing rate. A similar control is proven for the joint law of the exit time and and the exit position (where metastability means that they are approximately independent, with laws independent from the initial condition). As a consequence, a metastable behaviour is observed as soon as the spectral quantity is small with respect to the inverse of the absorbing rate. The results are tested for toy models (two or three states) in a low temperature regime.
scientific article; zbMATH DE number 7606004
scientific article