TY - INPR
A1 - Louis, Pierre-Yves
T1 - Increasing Coupling of Probabilistic Cellular Automata
N2 - 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.
T3 - Mathematische Statistik und Wahrscheinlichkeitstheorie : Preprint - 2004, 04
KW - stochastic ordering
KW - Probabilistic Cellular Automata
KW - monotone coupling
Y1 - 2004
U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-51578
ER -
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
U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-51560
ER -
TY - GEN
A1 - Louis, Pierre-Yves
T1 - Ergodicity of PCA
BT - equivalence between spatial and temporal mixing conditions
N2 - 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.
KW - Wahrscheinlichkeitstheorie
KW - Wechselwirkende Teilchensysteme
KW - Stochastische Zellulare Automaten
KW - Interacting particle systems
KW - Probabilistic Cellular Automata
KW - ERgodicity of Markov Chains
KW - Gibbs measures
Y1 - 2004
U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-6589
ER -
TY - GEN
A1 - Louis, Pierre-Yves
T1 - Increasing coupling for probabilistic cellular automata
N2 - 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.
KW - Wahrscheinlichkeitstheorie
KW - stochastische Anordnung
KW - stochastische Zellulare Automaten
KW - Kopplung
KW - stochastic ordering
KW - Probabilistic Cellular Automata
KW - monotone coupling
Y1 - 2005
U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-6593
ER -
TY - GEN
A1 - Dai Pra, Paolo
A1 - Louis, Pierre-Yves
A1 - Minelli, Ida
T1 - Monotonicity and complete monotonicity for continuous-time Markov chains
N2 - We analyze the notions of monotonicity and complete monotonicity for Markov Chains in continuous-time, taking values in a finite partially ordered set. Similarly to what happens in discrete-time, the two notions are not equivalent. However, we show that there are partially ordered sets for which monotonicity and complete monotonicity coincide in continuous time but not in discrete-time.
N2 - Nous étudions les notions de monotonie et de monotonie complète pour les processus de Markov (ou chaînes de Markov à temps continu) prenant leurs valeurs dans un espace partiellement ordonné. Ces deux notions ne sont pas équivalentes, comme c'est le cas lorsque le temps est discret. Cependant, nous établissons que pour certains ensembles partiellement ordonnés, l'équivalence a lieu en temps continu bien que n'étant pas vraie en temps discret.
KW - Stochastik
KW - continuous time Markov Chains
KW - poset
KW - monotonicity
KW - coupling
Y1 - 2006
U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-7665
ER -
TY - INPR
A1 - Pra, Paolo Dai
A1 - Louis, Pierre-Yves
A1 - Minelli, Ida G.
T1 - Complete monotone coupling for Markov processes
N2 - We formalize and analyze the notions of monotonicity and complete monotonicity for Markov Chains in continuous-time, taking values in a finite partially ordered set. Similarly to what happens in discrete-time, the two notions are not equivalent. However, we show that there are partially ordered sets for which monotonicity and complete monotonicity coincide in continuoustime but not in discrete-time.
T3 - Mathematische Statistik und Wahrscheinlichkeitstheorie : Preprint - 2008, 01
KW - Markov processes
KW - coupling
KW - partial ordering
KW - monotonicity conditions
KW - monotone random
KW - dynamical system representation
Y1 - 2008
U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-18286
ER -
TY - INPR
A1 - Louis, Pierre-Yves
A1 - Rouquier, Jean-Baptiste
T1 - Time-to-Coalescence for interacting particle systems : parallel versus sequential updating
N2 - Studying the influence of the updating scheme for MCMC algorithm on spatially extended models is a well known problem. For discrete-time interacting particle systems we study through simulations the effectiveness of a synchronous updating scheme versus the usual sequential one. We compare the speed of convergence of the associated Markov chains from the point of view of the time-to-coalescence arising in the coupling-from-the-past algorithm. Unlike the intuition, the synchronous updating scheme is not always the best one. The distribution of the time-to-coalescence for these spatially extended models is studied too.
T3 - Mathematische Statistik und Wahrscheinlichkeitstheorie : Preprint - 2009, 03
Y1 - 2009
U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-49454
ER -