Ergodic properties of Poissonian ID processes

Roy, Emmanuel

doi:10.1214/009117906000000692

Cited by 50 publications

(46 citation statements)

References 16 publications

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“…The first and fourth points are apparently due to Marchat in his PhD dissertation as pointed out by Grabinski in [8]. We have given our proof of these facts in [18]. The sixth point is folklore.…”

Section: T Is Of Zero Type Iff T * Is Mixing (And Then Mixing Of Any mentioning

confidence: 72%

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Poisson suspensions and infinite ergodic theory

Roy

2009

Ergod. Th. Dynam. Sys.

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Abstract. We investigate ergodic theory of Poisson suspensions. In the process, we establish close connections between finite and infinite measure preserving ergodic theory. Poisson suspensions thus provide a new approach to infinite measure ergodic theory. Fields investigated here are mixing properties, spectral theory, joinings. We also compare Poisson suspensions to the apparently similar looking Gaussian dynamical systems.

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Section: T Is Of Zero Type Iff T * Is Mixing (And Then Mixing Of Any mentioning

confidence: 72%

“…IDp) (this includes, in particular all α-stable stationary processes). We recall that these processes are exactly those that can be obtained as stochastic integrals with respect to a Poisson measure (see [13] and [18]). Theorem 5.6.…”

Section: 3mentioning

confidence: 99%

Poisson suspensions and infinite ergodic theory

Roy

2009

Ergod. Th. Dynam. Sys.

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“…It follows from (ii) that the Poisson suspension of T , which is a probability preserving transformation, is mixing of all orders [Roy1] with a simple spectrum [Ne]. For the definition of Poisson suspensions we refer to [CFS] and [Ne].…”

Section: Discussionmentioning

confidence: 99%

Spectral multiplicities of infinite measure preserving transformations

Danilenko

Ryzhikov

2010

Funct Anal Its Appl

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Abstract. Each subset E ⊂ N is realized as the set of essential values of the multiplicity function for the Koopman operator of an ergodic conservative infinite measure preserving transformation. IntroductionLet T be an ergodic conservative invertible measure preserving transformation of a σ-finite standard measure space (X, B, µ). Consider an associated unitary (Koopman) operator U T in the Hilbert space L 2 (X, µ):In the case of finite measure µ, the operator U T is usually considered only in the orthocomplement to the subspace of constant functions. A general question of the spectral theory of dynamical systems is (0-1) to find out which unitary operators can be realized as Koopman operators.The crux of the problem is related to the multiplicative property In the present paper we consider (0-2) in the class of infinite measure preserving conservative ergodic (i.e. type II ∞ ) transformations. It turns out that (0-2) can be solved completely in this class.

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“…However, it has been noticed in [17] that α-semi-stable stationary processes (α < 2) are factors of Poisson suspensions built over squashable systems (completely squashable in the stable case), associated with the Lévy measure of the process. Hence these Poisson suspensions are of zero or infinite entropy.…”

Section: Additivity and Scaling Of Poisson Entropymentioning

confidence: 99%

Poisson suspensions and entropy for infinite transformations

Janvresse

Meyerovitch

Roy

et al. 2009

Trans. Amer. Math. Soc.

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Abstract. The Poisson entropy of an infinite-measure-preserving transformation is defined in the 2005 thesis of Roy as the Kolmogorov entropy of its Poisson suspension. In this article, we relate Poisson entropy with other definitions of entropy for infinite transformations: For quasi-finite transformations we prove that Poisson entropy coincides with Krengel's and Parry's entropy. In particular, this implies that for null-recurrent Markov chains, the usual formula for the entropy, − q i p i,j log p i,j , holds for any definitions of entropy. Poisson entropy dominates Parry's entropy in any conservative transformation. We also prove that relative entropy (in the sense of Danilenko and Rudolph) coincides with the relative Poisson entropy. Thus, for any factor of a conservative transformation, difference of the Krengel's entropies equals difference of the Poisson entropies. In case there already exists a factor with zero Poisson entropy, we prove the existence of a maximum (Pinsker) factor with zero Poisson entropy. Together with the preceding results, this answers affirmatively the question raised by Aaronson and Park about existence of a Pinsker factor in the sense of Krengel for quasi-finite transformations.

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Ergodic properties of Poissonian ID processes

Cited by 50 publications

References 16 publications

Poisson suspensions and infinite ergodic theory

Poisson suspensions and infinite ergodic theory

Spectral multiplicities of infinite measure preserving transformations

Poisson suspensions and entropy for infinite transformations

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