4872
2004
eng
preprint
0
20110329


Coupling, space and time Mixing for parallel stochastic dynamics
We first introduce some coupling of a finite number of Probabilistic Cellular Automata dynamics (PCA), preserving the stochastic ordering. Using this tool, for a general attractive probabilistic cellular automata on SZd, where S is finite, we prove that a condition (A) is equivalent to the (time) convergence towards equilibrium of this Markovian parallel dynamics, in the uniform norm, exponentially fast. This condition (A) means the exponential decay of the influence from the boundary for the invariant measures of the system restricted to finite ‘box’volume. For a class of reversible PCA dynamics on {−1, +1}Zd , with a naturally associated Gibbsian potential ϕ, we prove that a Weak Mixing condition for ϕ implies the validity of the assumption (A); thus the ‘exponential ergodicity’ of the dynamics towards the unique Gibbs measure associated to ϕ holds. On some particular examples of this PCA class, we verify that our assumption (A) is weaker than the DobrushinVasershtein ergodicity condition. For some special PCA, the ‘exponential ergodicity’ holds as soon as there is no phase transition.
urn:nbn:de:kobv:517opus51560
5156
SI 990

Keine Nutzungslizenz vergeben  es gilt das deutsche Urheberrecht
PierreYves Louis
Mathematische Statistik und Wahrscheinlichkeitstheorie : Preprint
2004, 02
eng
uncontrolled
Probabilistic Cellular Automata
eng
uncontrolled
Interacting Particle Systems
eng
uncontrolled
Coupling
eng
uncontrolled
Attractive Dynamics
eng
uncontrolled
Stochastic Ordering
eng
uncontrolled
Weak Mixing Condition
Mathematik
open_access
Institut für Mathematik
Universität Potsdam
https://publishup.unipotsdam.de/files/4872/Preprint_2004_02.pdf