TY - JOUR 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 Y1 - 2006 UR - http://www.elsevier.com/locate/issn/1631073X U6 - https://doi.org/10.1016/j.crma.2006.04.007 SN - 1631-073X ER - TY - JOUR 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 greater as 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. Y1 - 2005 UR - http://www.sciencedirect.com/ science?_ob=JournalURL&_cdi=5672&_auth=y&_acct=C000053886&_version=1&_urlVersion=0&_userid=1584062&md5=a190494148436adc6a 89f14409e6e76e ER - TY - JOUR A1 - Louis, Pierre-Yves T1 - Ergodicity of PCA: equivalence between spatial and temporal mixing conditions N2 - 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. Y1 - 2004 UR - http://www.math.washington.edu/%7Eejpecp/ECP/viewarticle.php?id=1704&layout=abstract 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 - 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 - 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 - 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 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 - 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 -