scholarly journals Hausdorff spectrum of harmonic measure

2015 ◽  
Vol 37 (1) ◽  
pp. 277-307 ◽  
Author(s):  
RYOKICHI TANAKA
Keyword(s):  
Random Walk ◽  
Harmonic Measure ◽  
Hausdorff Measure ◽  
Hyperbolic Group ◽  
Volume Growth ◽  
Multifractal Spectrum ◽  
Parameter Family ◽  
Probability Measures ◽  
Dimensional Properties

For every non-elementary hyperbolic group, we show that for every random walk with finitely supported admissible step distribution, the associated entropy equals the drift times the logarithmic volume growth if and only if the corresponding harmonic measure is comparable with Hausdorff measure on the boundary. Moreover, we introduce one parameter family of probability measures which interpolates a Patterson–Sullivan measure and the harmonic measure, and establish a formula of Hausdorff spectrum (multifractal spectrum) of the harmonic measure. We also give some finitary versions of dimensional properties of the harmonic measure.

scholarly journals The ratio set of the harmonic measure of a random walk on a hyperbolic group

2008 ◽  
Vol 163 (1) ◽  
pp. 285-316 ◽  
Author(s):  
Masaki Izumi ◽  
Sergey Neshveyev ◽  
Rui Okayasu
Keyword(s):  
Random Walk ◽  
Harmonic Measure ◽  
Hyperbolic Group ◽  
Ratio Set

INTERPOLATION OF CHEBYSHEV POLYNOMIALS AND INTERACTING FOCK SPACES

2006 ◽  
Vol 09 (03) ◽  
pp. 361-371 ◽  
Author(s):  
IZUMI KUBO ◽  
HUI-HSIUNG KUO ◽  
SUAT NAMLI
Keyword(s):  
Orthogonal Polynomials ◽  
Anderson Model ◽  
Chebyshev Polynomials ◽  
Field Operator ◽  
Parameter Family ◽  
Probability Measures ◽  
Fock Spaces ◽  
Arcsine Distribution ◽  
Multiplicative Renormalization ◽  
Renormalization Method

We discover a family of probability measures μa, 0 < a ≤ 1, [Formula: see text] which contains the arcsine distribution (a = 1) and semi-circle distribution (a = 1/2). We show that the multiplicative renormalization method can be used to produce orthogonal polynomials, called Chebyshev polynomials with one parameter a, which reduce to Chebyshev polynomials of the first and second kinds when a = 1 and 1/2 respectively. Moreover, we derive the associated Jacobi–Szegö parameters. This one-parameter family of probability measures coincides with the vacuum distribution of the field operator of the interacting Fock spaces related to the Anderson model.

scholarly journals Relative multifractal box-dimensions

Filomat ◽  
2019 ◽  
Vol 33 (9) ◽  
pp. 2841-2859 ◽  
Author(s):  
Najmeddine Attia ◽  
Bilel Selmi
Keyword(s):  
Hausdorff Dimension ◽  
Multifractal Spectrum ◽  
Packing Dimension ◽  
Probability Measures ◽  
Multifractal Formalism ◽  
The Relationship ◽  
Box Dimensions

Given two probability measures ? and ? on Rn. We define the upper and lower relative multifractal box-dimensions of the measure ? with respect to the measure ? and investigate the relationship between the multifractal box-dimensions and the relative multifractal Hausdorff dimension, the relative multifractal pre-packing dimension. We also, calculate the relative multifractal spectrum and establish the validity of multifractal formalism. As an application, we study the behavior of projections of measures obeying to the relative multifractal formalism.

scholarly journals Harmonic measure is rectifiable if it is absolutely continuous with respect to the co-dimension-one Hausdorff measure

2016 ◽  
Vol 354 (4) ◽  
pp. 351-355
Author(s):  
Jonas Azzam ◽  
Steve Hofmann ◽  
José María Martell ◽  
Svitlana Mayboroda ◽  
Mihalis Mourgoglou ◽  
...  
Keyword(s):  
Harmonic Measure ◽  
Hausdorff Measure ◽  
Absolutely Continuous ◽  
Dimension One

Ergodic theory on Galton—Watson trees: speed of random walk and dimension of harmonic measure

1995 ◽  
Vol 15 (3) ◽  
pp. 593-619 ◽  
Author(s):  
Russell Lyons ◽  
Robin Pemantle ◽  
Yuval Peres
Keyword(s):  
Random Walk ◽  
Ergodic Theory ◽  
Harmonic Measure ◽  
Branching Process ◽  
Main Tool ◽  
Simple Random Walk ◽  
Family Tree ◽  
First Order ◽  
The Family ◽  
Rate Of Escape

AbstractWe consider simple random walk on the family treeTof a nondegenerate supercritical Galton—Watson branching process and show that the resulting harmonic measure has a.s. strictly smaller Hausdorff dimension than that of the whole boundary ofT. Concretely, this implies that an exponentially small fraction of thenth level ofTcarries most of the harmonic measure. First-order asymptotics for the rate of escape, Green function and the Avez entropy of the random walk are also determined. Ergodic theory of the shift on the space of random walk paths on trees is the main tool; the key observation is that iterating the transformation induced from this shift to the subset of ‘exit points’ yields a nonintersecting path sampled from harmonic measure.

A Discrete Analogue of a Theorem of Makarov

1993 ◽  
Vol 2 (2) ◽  
pp. 181-199 ◽  
Author(s):  
Gregory F. Lawler
Keyword(s):  
Random Walk ◽  
Hausdorff Dimension ◽  
Harmonic Measure ◽  
Simple Random Walk ◽  
Discrete Analogue ◽  
Connected Subset ◽  
Image Position ◽  
Dimension One

A theorem of Makarov states that the harmonic measure of a connected subset of ℝ2 is supported on a set of Hausdorff dimension one. This paper gives an analogue of this theorem for discrete harmonic measure, i.e., the hitting measure of simple random walk. It is proved that for any 1/2 < α < 1, β < α − 1/2, there is a constant k such that for any connected subset A ⊂ ℤ2 of radius n,where HA denotes discrete harmonic measure.

scholarly journals A note on the geometric ergodicity of a Markov chain

1989 ◽  
Vol 21 (3) ◽  
pp. 702-704 ◽  
Author(s):  
K. S. Chan
Keyword(s):  
Markov Chain ◽  
Random Walk ◽  
Convergence Rates ◽  
Transition Probability ◽  
Invariant Probability Measure ◽  
Autoregressive Process ◽  
Probability Measures ◽  
Geometric Ergodicity ◽  
Drift Condition ◽  
Step Transition

It is known that if an irreducible and aperiodic Markov chain satisfies a ‘drift' condition in terms of a non-negative measurable function g(x), it is geometrically ergodic. See, e.g. Nummelin (1984), p. 90. We extend the analysis to show that the distance between the nth-step transition probability and the invariant probability measure is bounded above by ρ n(a + bg(x)) for some constants a, b> 0 and ρ < 1. The result is then applied to obtain convergence rates to the invariant probability measures for an autoregressive process and a random walk on a half line.

scholarly journals Amenability of groupoids and asymptotic invariance of convolution powers

2021 ◽  
pp. 69-92
Author(s):  
Theo Bühler ◽  
Vadim Kaimanovich
Keyword(s):  
Random Walk ◽  
Locally Compact Group ◽  
Probability Measures ◽  
Locally Compact ◽  
Von Neumann ◽  
Original Definition ◽  
Convolution Powers ◽  
Measure Class ◽  
Definition Of

The original definition of amenability given by von Neumann in the highly non-constructive terms of means was later recast by Day using approximately invariant probability measures. Moreover, as it was conjectured by Furstenberg and proved by Kaimanovich–Vershik and Rosenblatt, the amenability of a locally compact group is actually equivalent to the existence of a single probability measure on the group with the property that the sequence of its convolution powers is asymptotically invariant. In the present article we extend this characterization of amenability to measured groupoids. It implies, in particular, that the amenability of a measure class preserving group action is equivalent to the existence of a random environment on the group parameterized by the action space, and such that the tail of the random walk in almost every environment is trivial.

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