Stochastic dynamical systems with weak contractivity properties I. Strong and local contractivity

Peigné, Marc; Woess, Wolfgang

doi:10.4064/cm125-1-4

Cited by 25 publications

(68 citation statements)

References 25 publications

(30 reference statements)

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“…The process has been also studied in more general settings when u n admits also negative values (see Peigné, Woess [20] for recent results and a comprehensive bibliography).…”

Section: 4mentioning

confidence: 99%

“…Peigné and Woess [20] proved that if E(u + 1 ) 3/2 < ∞, for u + 1 = max{u 1 , 0}, then the process {X n } is recurrent on R + . As a consequence of Benda's theorem, the process possesses a unique invariant Radon measure ν on R + (local contractivity is easy to prove).…”

Section: 4mentioning

confidence: 99%

“…Less is known for the null recurrent case, especially in a general setting. Existence and uniqueness of an invariant Radon measure have been the topic of two recent works: Deroin et al [8] on symmetric SDS of homeomorphism of R, and Peigné and Woess [20] on the phenomenon of local contraction. We refer to them for a more complete bibliography on the subject.…”

Section: 1mentioning

confidence: 99%

“…He did not publish his results, however, they have been recently incorporated, with a complete and simplified proof, into two papers of Peigné and Woess [20,21], where they also investigated ergodicity of SDS generated by Lipschitz maps with centered Lipschitz's coefficient.…”

mentioning

confidence: 99%

“…In view of the result of Chow and Lai [7], E(ū k ) 1/2 < ∞ and this is sufficient for existence of a unique invariant probability measure ν t of the process {Y x k } (see [20] for more details). Let us define the measure…”

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confidence: 99%

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On unbounded invariant measures of stochastic dynamical systems

Brofferio¹,

Buraczewski²

2015

Ann. Probab.

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We consider stochastic dynamical systems on R, that is, random processes defined by X x n = Ψn(X x n−1 ), X x 0 = x, where Ψn are i.i.d. random continuous transformations of some unbounded closed subset of R. We assume here that Ψn behaves asymptotically like Anx, for some random positive number An [the main example is the affine stochastic recursion Ψn(x) = Anx + Bn]. Our aim is to describe invariant Radon measures of the process X x n in the critical case, when E log A1 = 0. We prove that those measures behave at infinity like dx x . We study also the problem of uniqueness of the invariant measure. We improve previous results known for the affine recursions and generalize them to a larger class of stochastic dynamical systems which include, for instance, reflected random walks, stochastic dynamical systems on the unit interval [0, 1], additive Markov processes and a variant of the Galton-Watson process.

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“…The process has been also studied in more general settings when u n admits also negative values (see Peigné, Woess [20] for recent results and a comprehensive bibliography).…”

Section: 4mentioning

confidence: 99%

Section: 4mentioning

confidence: 99%

Section: 1mentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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On unbounded invariant measures of stochastic dynamical systems

Brofferio¹,

Buraczewski²

2015

Ann. Probab.

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Limit Theorem for Reflected Random Walks

Ngo

Peigné

2021

Lecture Notes in Mathematics

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HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L'archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

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On Solutions of the Affine Recursion and the Smoothing Transform in the Critical Case

Brofferio

Buraczewski

Damek

2013

Springer Proceedings in Mathematics &Amp; Statistics

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We fix d ≥ 2 and denote S the semi-group of d × d matrices with non negative entries. We consider a sequence (A n , B n ) n≥1 of i. i. d. random variables with values in S × R d + and study the asymptotic behavior of the Markov chain (X n ) n≥0 on R d + defined by:where X 0 is a fixed random variable. We assume that the Lyapunov exponent of the matrices A n equals 0 and prove, under quite general hypotheses, that there exists a unique (infinite) Radon measure λ on (R + ) d which is invariant for the chain (X n ) n≥0 . The existence of λ relies on a recent work by T.D.C. Pham about fluctuations of the norm of product of random matrices [16]. Its unicity is a consequence of a general property, called "local contractivity", highlighted about 20 years ago by M. Babillot, Ph. Bougerol et L. Elie in the case of the one dimensional affine recursion [1] .

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Stochastic dynamical systems with weak contractivity properties I. Strong and local contractivity

Cited by 25 publications

References 25 publications

On unbounded invariant measures of stochastic dynamical systems

On unbounded invariant measures of stochastic dynamical systems

Limit Theorem for Reflected Random Walks

On Solutions of the Affine Recursion and the Smoothing Transform in the Critical Case

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