##
**Metastable behavior of stochastic dynamics: A pathwise approach.**
*(English)*
Zbl 0591.60080

Summary: A new approach to metastability for stochastic dynamics is proposed. The basic idea is to study the statistics of each path, performing time averages along the evolution. Metastability would be characterized by the fact that the process of these time averages converges, under a suitable rescaling, to a measure valued Markov jump process. Here this convergence is shown for the Curie-Weiss mean field dynamics and also for a model with spatial structure: Harris contact process.

### MSC:

60J70 | Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) |

60J75 | Jump processes (MSC2010) |

60K35 | Interacting random processes; statistical mechanics type models; percolation theory |

70F99 | Dynamics of a system of particles, including celestial mechanics |

### Keywords:

metastability; mean field theory; contact process; measure valued Markov jump process; Curie-Weiss mean field dynamics; Harris contact process
PDF
BibTeX
XML
Cite

\textit{M. Cassandro} et al., J. Stat. Phys. 35, 603--634 (1984; Zbl 0591.60080)

Full Text:
DOI

### References:

[1] | P. Billingsley,Convergence of Probability Measure (John Wiley and Sons, New York, 1968). · Zbl 0172.21201 |

[2] | D. Capocaccia, M. Cassandro, and E. Olivieri, A study of metastability in the Ising model,Commun. Math. Physics 39:185-205 (1974). |

[3] | R. Durret and D. Griffeath, Supercritical contact processes on ?d,Ann. Probability 11:1-15 (1983). · Zbl 0508.60080 |

[4] | W. Fleming and M. Viot, Some measure-valued Markov processes in population genetics,Indiana Univ. Math. J. 28:817 (1979). · Zbl 0444.60064 |

[5] | D. Griffeath,Additive and Cancellative Interacting Particle Systems (Lecture Notes in Mathematics, No. 724. Springer, New York, 1979). · Zbl 0412.60095 |

[6] | D. Griffeath, The basic contact processes,Stochastic Processes and Their Applications 11:151-185 (1981). · Zbl 0463.60085 |

[7] | R. Griffiths, C. Y. Weng, and J. S. Langer, Relaxation times for metastable states in the mean-field model of a ferromagnet,Phys. Rev. 149:301-305 (1966). |

[8] | T. E. Harris, Contact interactions on a lattice,Ann. Probability 2:969-988 (1974). · Zbl 0334.60052 |

[9] | T. E. Harris, Additive set valued Markov processes and graphical methods,Ann. Probability 6:355-378 (1978). · Zbl 0378.60106 |

[10] | J. L. Lebowitz and O. Penrose, Rigorous treatment of the Van der Walls-Maxwell theory of liquid-vapor transition,J. Math. Phys. 7:98-113 (1966). · Zbl 0938.82520 |

[11] | J. L. Lebowitz and O. Penrose, Rigorous treatment of metastable states in the Van der Walls-Maxwell theory,J. Stat. Phys. 3:211-236 (1971). · Zbl 0938.82521 |

[12] | A. D. Ventsel and M. L. Freidlin, On small random perturbations of dynamical systems,Russ. Math. Surv. 25:1-55 (1970). · Zbl 0297.34053 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.