Ergodic properties of smooth hyperbolc measures

2001 ◽  
pp. 115-144
Author(s):  
Luis Barreira ◽  
Yakov Pesin
Keyword(s):  
Ergodic Properties

scholarly journals Isomorphisms between determinantal point processes with translation-invariant kernels and Poisson point processes

2020 ◽  
pp. 1-14
Author(s):  
SHOTA OSADA
Keyword(s):  
Point Processes ◽  
Duke Math ◽  
Random Point ◽  
Determinantal Point Processes ◽  
Fredholm Determinants ◽  
Poisson Point Processes ◽  
Translation Invariant ◽  
Ergodic Properties ◽  
Tree Representations ◽  
Invariant Kernels

Abstract We prove the Bernoulli property for determinantal point processes on $ \mathbb{R}^d $ with translation-invariant kernels. For the determinantal point processes on $ \mathbb{Z}^d $ with translation-invariant kernels, the Bernoulli property was proved by Lyons and Steif [Stationary determinantal processes: phase multiplicity, bernoullicity, and domination. Duke Math. J.120 (2003), 515–575] and Shirai and Takahashi [Random point fields associated with certain Fredholm determinants II: fermion shifts and their ergodic properties. Ann. Probab.31 (2003), 1533–1564]. We prove its continuum version. For this purpose, we also prove the Bernoulli property for the tree representations of the determinantal point processes.

scholarly journals Ergodic properties of classical dissipative systems I

1998 ◽  
Vol 181 (2) ◽  
pp. 245-282 ◽  
Author(s):  
Vojkan Jakšić ◽  
Claude-Alain Pillet
Keyword(s):  
Dissipative Systems ◽  
Ergodic Properties

Numerical study on ergodic properties of triangular billiards

1997 ◽  
Vol 55 (6) ◽  
pp. 6384-6390 ◽  
Author(s):  
Roberto Artuso ◽  
Giulio Casati ◽  
Italo Guarneri
Keyword(s):  
Numerical Study ◽  
Ergodic Properties

Markov paths on the Poisson-Delaunay graph with applications to routeing in mobile networks

2000 ◽  
Vol 32 (01) ◽  
pp. 1-18 ◽  
Author(s):  
F. Baccelli ◽  
K. Tchoumatchenko ◽  
S. Zuyev
Keyword(s):  
Communication Networks ◽  
Mobile Networks ◽  
Path Length ◽  
Euclidean Distance ◽  
Voronoi Tessellation ◽  
Voronoi Cells ◽  
Ergodic Properties ◽  
Potential Applications ◽  
The Mean ◽  
Mobile Communication Networks

Consider the Delaunay graph and the Voronoi tessellation constructed with respect to a Poisson point process. The sequence of nuclei of the Voronoi cells that are crossed by a line defines a path on the Delaunay graph. We show that the evolution of this path is governed by a Markov chain. We study the ergodic properties of the chain and find its stationary distribution. As a corollary, we obtain the ratio of the mean path length to the Euclidean distance between the end points, and hence a bound for the mean asymptotic length of the shortest path. We apply these results to define a family of simple incremental algorithms for constructing short paths on the Delaunay graph and discuss potential applications to routeing in mobile communication networks.

scholarly journals Ergodic properties of quantized toral automorphisms

1997 ◽  
Vol 38 (1) ◽  
pp. 67-83 ◽  
Author(s):  
Sławomir Klimek ◽  
Andrzej Leśniewski ◽  
Neepa Maitra ◽  
Ron Rubin
Keyword(s):  
Ergodic Properties ◽  
Toral Automorphisms

scholarly journals PAC-Bayes Bounds on Variational Tempered Posteriors for Markov Models

Entropy ◽  
2021 ◽  
Vol 23 (3) ◽  
pp. 313
Author(s):  
Imon Banerjee ◽  
Vinayak A. Rao ◽  
Harsha Honnappa
Keyword(s):  
Markov Model ◽  
Markov Models ◽  
Model Misspecification ◽  
Dependence Structure ◽  
Posterior Distributions ◽  
Ergodic Properties ◽  
Variational Approximations ◽  
Popular View ◽  
Probably Approximately Correct ◽  
Risk Bounds

Datasets displaying temporal dependencies abound in science and engineering applications, with Markov models representing a simplified and popular view of the temporal dependence structure. In this paper, we consider Bayesian settings that place prior distributions over the parameters of the transition kernel of a Markov model, and seek to characterize the resulting, typically intractable, posterior distributions. We present a Probably Approximately Correct (PAC)-Bayesian analysis of variational Bayes (VB) approximations to tempered Bayesian posterior distributions, bounding the model risk of the VB approximations. Tempered posteriors are known to be robust to model misspecification, and their variational approximations do not suffer the usual problems of over confident approximations. Our results tie the risk bounds to the mixing and ergodic properties of the Markov data generating model. We illustrate the PAC-Bayes bounds through a number of example Markov models, and also consider the situation where the Markov model is misspecified.

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