[Cut-off et sortie de la métastabilité : les deux faces de la même pièce]
Nous présentons un cadre général qui relie cut-off et excursions de sortie pour des processus de naissance et de mort sur un alphabet dénombrable. Sous des hypothèses adaptées, nous montrons que le cut-off vers un équilibre (local) est accompagné par une distribution exponentielle des temps de sortie de l'équilibre. De plus, les trajectoires atypiques menant à ces excursions sont les renversées temporelles de trajectoires de cut-off ; en particulier leurs durées suivent la même loi.
We present a general framework linking cut-off and exit excursions for birth-and-death processes on a countable alphabet. Under suitable hypotheses, we prove that cut-off convergence towards a (local) equilibrium is accompanied by exponentially distributed out-of-equilibrium excursions. Furthermore, atypical trajectories leading to these excursions and final cut-off trajectories are related by time inversion; in particular their time lengths have identical laws.
Accepté le :
Publié le :
@article{CRMATH_2008__346_11-12_691_0,
author = {Olivier Bertoncini and Javiera Barrera M. and Roberto Fern\'andez},
title = {Cut-off and exit from metastability: two sides of the same coin},
journal = {Comptes Rendus. Math\'ematique},
pages = {691--696},
publisher = {Elsevier},
volume = {346},
number = {11-12},
year = {2008},
doi = {10.1016/j.crma.2008.04.007},
language = {en},
}
TY - JOUR AU - Olivier Bertoncini AU - Javiera Barrera M. AU - Roberto Fernández TI - Cut-off and exit from metastability: two sides of the same coin JO - Comptes Rendus. Mathématique PY - 2008 SP - 691 EP - 696 VL - 346 IS - 11-12 PB - Elsevier DO - 10.1016/j.crma.2008.04.007 LA - en ID - CRMATH_2008__346_11-12_691_0 ER -
Olivier Bertoncini; Javiera Barrera M.; Roberto Fernández. Cut-off and exit from metastability: two sides of the same coin. Comptes Rendus. Mathématique, Volume 346 (2008) no. 11-12, pp. 691-696. doi : 10.1016/j.crma.2008.04.007. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2008.04.007/
[1] Random walks on finite groups and rapidly mixing Markov chains, Seminar on Probability, XVII, Lecture Notes in Math., vol. 986, Springer, Berlin, 1983, pp. 243-297
[2] Shuffling cards and stopping times, Amer. Math. Monthly, Volume 93 (1986) no. 5, pp. 333-348
[3] Strong uniform times and finite random walks, Adv. Appl. Math., Volume 8 (1987) no. 1, pp. 69-97
[4] J. Barrera M., O. Bertoncini, R. Fernández, Abrupt convergence and metastability for birth and death chains, in preparation
[5] O. Bertoncini, Convergence abrupte et métastabilité, Thesis, Université de Rouen, 2007
[6] Metastability in stochastic dynamics of disordered mean-field models, Probab. Theory Related Fields, Volume 119 (2001) no. 1, pp. 99-161
[7] Metastability and low lying spectra in reversible Markov chains, Commun. Math. Phys., Volume 228 (2002) no. 2, pp. 219-255
[8] Metastable behavior of stochastic dynamics: a pathwise approach, J. Statist. Phys., Volume 35 (1984) no. 5–6, pp. 603-634
[9] Group Representations in Probability and Statistics, Institute of Mathematical Statistics Lecture Notes—Monograph Series, vol. 11, Institute of Mathematical Statistics, Hayward, CA, 1988
[10] The cutoff phenomenon in finite Markov chains, Proc. Natl. Acad. Sci. USA, Volume 93 (1996) no. 4, pp. 1659-1664
[11] Decay rates and cutoff for convergence and hitting times of Markov chains with countably infinite state space, Adv. Appl. Probab., Volume 33 (2001) no. 1, pp. 188-205
[12] Markov chains with exponentially small transition probabilities: first exit problem from a general domain. I. The reversible case, J. Statist. Phys., Volume 79 (1995) no. 3–4, pp. 613-647
[13] Markov chains with exponentially small transition probabilities: first exit problem from a general domain. II. The general case, J. Statist. Phys., Volume 84 (1996) no. 5–6, pp. 987-1041
[14] Large Deviations and Metastability, Encyclopedia of Mathematics and its Applications, vol. 100, Cambridge University Press, Cambridge, 2005
[15] Lectures on finite Markov chains, Saint-Flour, 1996 (Lecture Notes in Math.), Volume vol. 1665, Springer, Berlin (1997), pp. 301-413
[16] Random walks on finite groups, Probability on Discrete Structures, Encyclopaedia Math. Sci., vol. 110, Springer, Berlin, 2004, pp. 263-346
[17] The pattern of escape from metastability of a stochastic Ising model, Commun. Math. Phys., Volume 147 (1992) no. 2, pp. 231-240
[18] Metastability for Markov chains: a general procedure based on renormalization group ideas, Cambridge, 1993 (NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci.), Volume vol. 420, Kluwer Acad. Publ., Dordrecht (1994), pp. 303-322
[19] Cutoff for Markov chains: some examples and applications, Santiago, 1998 (Nonlinear Phenom. Complex Systems), Volume vol. 6, Kluwer Acad. Publ., Dordrecht (2001), pp. 261-300
[20] Modèles et algorithmes markoviens, Mathématiques & Applications (Berlin), vol. 39, Springer-Verlag, Berlin, 2002
Cité par Sources :
Commentaires - Politique
Étude théorique de la compression de spin nucléaire par mesure quantique non destructive en continu
Alan Serafin; Yvan Castin; Matteo Fadel; ...
C. R. Phys (2021)
Densité des zéros des transformés de Lévy itérés d'un mouvement brownien
Marc Malric
C. R. Math (2003)