TY - INPR
A1 - Louis, Pierre-Yves
T1 - Coupling, space and time Mixing for parallel stochastic dynamics
N2 - 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 Dobrushin-Vasershtein ergodicity condition. For some special PCA, the ‘exponential ergodicity’ holds as soon as there is no phase transition.
T3 - Mathematische Statistik und Wahrscheinlichkeitstheorie : Preprint - 2004, 02
KW - Probabilistic Cellular Automata
KW - Interacting Particle Systems
KW - Coupling
KW - Attractive Dynamics
KW - Stochastic Ordering
KW - Weak Mixing Condition
Y1 - 2004
UR - https://publishup.uni-potsdam.de/frontdoor/index/index/docId/4872
UR - https://nbn-resolving.org/urn:nbn:de:kobv:517-opus-51560
ER -