Processus des misanthropes. (French) Zbl 0554.60097

Cocozza-Thivent, Christiane

Z. Wahrscheinlichkeitstheor. Verw. Geb. 70, 509-523 (1985).

We consider a system of identical interacting particles moving on the lattice \({\mathbb{Z}}^ d\). The rate at which a particle at the site x jumps to the site y is p(y-x)b(\(\eta\) (x),\(\eta\) (y)) where p is an irreducible probability on \({\mathbb{Z}}^ d\) and b(\(\eta\) (x),\(\eta\) (y)) is an increasing (resp. decreasing) function of the number \(\eta\) (x) (resp. \(\eta\) (y)) of particles at site x (resp. y). We study the convergence of the system to equilibrium and describe the invariant measures.

MSC:

60K35	Interacting random processes; statistical mechanics type models; percolation theory
60B10	Convergence of probability measures

Keywords:

interacting particles; convergence of the system to equilibrium; invariant measures

References:

[1]	Andjel, E., Invariant measures for the zero range process, Ann. Prob., 10, 525-547 (1982) · Zbl 0492.60096
[2]	Andjel, E., The asymetric simple exclusion process on ℤ^d, Z. Wahrscheinlichkeitstheor. Verw. Geb., 58, 423-432 (1981) · Zbl 0458.60097
[3]	Cocozza, Ch.; Kipnis, C., Processus de vie et mort sur ℝ avec interaction selon les particules les plus proches, Z. Wahrscheinlichkeitstheor. Verw. Geb., 51, 123-132 (1980) · Zbl 0428.60090
[4]	Cocozza, Ch.; Roussignol, M., Unicité d’un processus de naissance et mort sur la droite réelle, Annales de l’IHP, 15, 93-105 (1979) · Zbl 0405.60082
[5]	Holley, R.; Stroock, D., A martingale approach to infinite systems of interacting processes, Ann. Probab., 4, 195-228 (1976) · Zbl 0332.60072
[6]	Holley, R.; Stroock, D., Nearest neighbor birth and death processes on the real line, Acta Math., 140, 103-154 (1978) · Zbl 0405.60090
[7]	Liggett, T., The stochastic evolution of infinite systems for interacting particles, Ecole d’Eté de probabilités de Saint-Flour VI, 1976 (1977), Berlin-Heidelberg-New York: Springer, Berlin-Heidelberg-New York · Zbl 0363.60109
[8]	Liggett, T.: Interacting particle systems. [à paraître chez Springer] · Zbl 0559.60078
[9]	Spitzer, F., Interacting Markov processes, Adv. Math., 5, 246-290 (1970) · Zbl 0312.60060