On quantitative convergence to quasi-stationarity

Diaconis, Persi; Miclo, Laurent

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 24 (2015) no. 4, pp. 973-1016.

On cherche à quantifier le comportement en temps long des processus de Markov finis, absorbés et supposés irréductibles en dehors du point absorbant. Par le biais de transformations de Doob, on montre qu’il est juste besoin du ratio maximal des valeurs prises par le premier vecteur propre de Dirichlet associé pour se ramener à la situation bien plus étudiée de la convergence à l’équilibre des processus de Markov finis. On obtient ainsi des estimées explicites de convergence à la quasi-stationnarité, en particulier via l’utilisation d’inégalités fonctionnelles. Quand le processus est de plus réversible, on retrouve le taux optimal de convergence exponentielle donné par le trou spectral entre les deux premières valeurs propres de Dirichlet. Plusieurs exemples simples illustrent les bornes obtenues.

The quantitative long time behavior of absorbing, finite, irreducible Markov processes is considered. Via Doob transforms, it is shown that only the knowledge of the ratio of the values of the underlying first Dirichlet eigenvector is necessary to come back to the well-investigated situation of the convergence to equilibrium of ergodic finite Markov processes. This leads to explicit estimates on the convergence to quasi-stationarity, in particular via functional inequalities. When the process is reversible, the optimal exponential rate consisting of the spectral gap between the two first Dirichlet eigenvalues is recovered. Several simple examples are provided to illustrate the bounds obtained.

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