@article{Louis2004,
  author    = {Louis, Pierre-Yves},
  title     = {Ergodicity of PCA: equivalence between spatial and temporal mixing conditions},
  year      = {2004},
  abstract  = {For a general attractive Probabilistic Cellular Automata on SZd, we prove that the (time-) convergence towards equilibrium of this Markovian parallel dynamics exponentially fast in the uniform norm is equivalent to a condition (A). This condition means the exponential decay of the influence from the boundary for the invariant measures of the system restricted to finite boxes. For a class of reversible PCA dynamics on {;1, +1}Zd, with a naturally associated Gibbsian potential ;, we prove that a (spatial-) weak mixing condition (WM) for ; implies the validity of the assumption (A); thus exponential (time-) ergodicity of these dynamics towards the unique Gibbs measure associated to ; holds. On some particular examples we state that exponential ergodicity holds as soon as there is no phase transition.},
  language  = {en}
}
@misc{Louis2004,
  author    = {Louis, Pierre-Yves},
  title     = {Ergodicity of PCA},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-6589},
  year      = {2004},
  abstract  = {For a general attractive Probabilistic Cellular Automata on S-Zd, we prove that the (time-) convergence towards equilibrium of this Markovian parallel dynamics, exponentially fast in the uniform norm, is equivalent to a condition (A). This condition means the exponential decay of the influence from the boundary for the invariant measures of the system restricted to finite boxes. For a class of reversible PCA dynamics on {1,+1}(Zd), wit a naturally associated Gibbsian potential rho, we prove that a (spatial-) weak mixing condition (WM) for rho implies the validity of the assumption (A); thus exponential (time-) ergodicity of these dynamics towards the unique Gibbs measure associated to rho hods. On some particular examples we state that exponential ergodicity holds as soon as there is no phase transition.},
  subject      = {Wahrscheinlichkeitstheorie},
  language  = {en}
}
@unpublished{Louis2004,
  author    = {Louis, Pierre-Yves},
  title     = {Coupling, space and time Mixing for parallel stochastic dynamics},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-51560},
  year      = {2004},
  abstract  = {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.},
  language  = {en}
}
@unpublished{Louis2004,
  author    = {Louis, Pierre-Yves},
  title     = {Increasing Coupling of Probabilistic Cellular Automata},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-51578},
  year      = {2004},
  abstract  = {We give a necessary and sufficient condition for the existence of an increasing coupling of N (N >= 2) synchronous dynamics on S-Zd (PCA). Increasing means the coupling preserves stochastic ordering. We first present our main construction theorem in the case where S is totally ordered; applications to attractive PCAs are given. When S is only partially ordered, we show on two examples that a coupling of more than two synchronous dynamics may not exist. We also prove an extension of our main result for a particular class of partially ordered spaces.},
  language  = {en}
}