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

scholarly journals The Mutual Singularity of Harmonic Measure and Hausdorff Measure of Codimension Smaller than One

2019 ◽  
Author(s):  
Xavier Tolsa
Keyword(s):  
Harmonic Measure ◽  
Hausdorff Measure ◽  
Additional Assumption ◽  
Uniform Domain ◽  
Density Condition ◽  
Absolutely Continuous ◽  
Local Capacity

Abstract Let $\Omega \subset{\mathbb{R}}^{n+1}$ be open and let $E\subset \partial \Omega $ with $0<{\mathcal{H}}^s(E)<\infty $, for some $s\in (n,n+1)$, satisfy a local capacity density condition. In this paper it is shown that the harmonic measure cannot be mutually absolutely continuous with the Hausdorff measure ${\mathcal{H}}^s$ on $E$. This answers a question of Azzam and Mourgoglou, who had proved the same result under the additional assumption that $\Omega $ is a uniform domain.

scholarly journals Local Dimensions of Measures of Finite Type II: Measures Without Full Support and With Non-regular Probabilities

2018 ◽  
Vol 70 (4) ◽  
pp. 824-867 ◽  
Author(s):  
Kathryn E. Hare ◽  
Kevin G. Hare ◽  
Michael Ka Shing Ng
Keyword(s):  
Finite Type ◽  
Hausdorff Measure ◽  
Finite Sequence ◽  
Pisot Number ◽  
Local Dimension ◽  
Bernoulli Convolution ◽  
Similar Measure ◽  
Absolutely Continuous ◽  
Contraction Ratio ◽  
Isolated Point

AbstractConsider a finite sequence of linear contractions Sj(x) = px + dj and probabilities pj > 0 with ∑Pj = 1. We are interested in the self-similar measure , of finite type. In this paper we study the multi-fractal analysis of such measures, extending the theory to measures arising from non-regular probabilities and whose support is not necessarily an interval.Under some mild technical assumptions, we prove that there exists a subset of supp μ of full μ and Hausdorff measure, called the truly essential class, for which the set of (upper or lower) local dimensions is a closed interval. Within the truly essential class we show that there exists a point with local dimension exactly equal to the dimension of the support. We give an example where the set of local dimensions is a two element set, with all the elements of the truly essential class giving the same local dimension. We give general criteria for these measures to be absolutely continuous with respect to the associated Hausdorff measure of their support, and we show that the dimension of the support can be computed using only information about the essential class.To conclude, we present a detailed study of three examples. First, we show that the set of local dimensions of the biased Bernoulli convolution with contraction ratio the inverse of a simple Pisot number always admits an isolated point. We give a precise description of the essential class of a generalized Cantor set of finite type, and show that the k-th convolution of the associated Cantor measure has local dimension at x ∊ (0,1) tending to 1 as ft: tends to infinity. Lastly, we show that within a maximal loop class that is not truly essential, the set of upper local dimensions need not be an interval. This is in contrast to the case for finite type measures with regular probabilities and full interval support.

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.

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 Necessary Condition for Rectifiability Involving Wasserstein Distance W2

2020 ◽  
Author(s):  
Damian Dąbrowski
Keyword(s):  
Hausdorff Measure ◽  
Radon Measure ◽  
Recent Article ◽  
Wasserstein Distance ◽  
Necessary Condition ◽  
Absolutely Continuous ◽  
Dimensional Hausdorff Measure ◽  
Almost All

Abstract A Radon measure $\mu $ is $n$-rectifiable if it is absolutely continuous with respect to $n$-dimensional Hausdorff measure and $\mu $-almost all of ${\operatorname{supp}}\mu $ can be covered by Lipschitz images of $\mathbb{R}^n$. In this paper, we give a necessary condition for rectifiability in terms of the so-called $\alpha _2$ numbers — coefficients quantifying flatness using Wasserstein distance $W_2$. In a recent article, we showed that the same condition is also sufficient for rectifiability, and so we get a new characterization of rectifiable measures.

Absolute Continuity of Wasserstein Barycenters Over Alexandrov Spaces

2017 ◽  
Vol 69 (5) ◽  
pp. 1087-1108
Author(s):  
Yin Jiang
Keyword(s):  
Hausdorff Measure ◽  
Absolute Continuity ◽  
Alexandrov Space ◽  
Probability Measures ◽  
Absolutely Continuous ◽  
Alexandrov Spaces ◽  
Dimensional Hausdorff Measure ◽  
Borel Probability

AbstractIn this paper, we prove that on a compact, n-dimensional Alexandrov space with curvature at least −1, the Wasserstein barycenter of Borel probability measures μ1 ,… , μm is absolutely continuous with respect to the n-dimensional Hausdorff measure if one of them is.

DERIVATION BASES AND THE HAUSDORFF MEASURE

1986 ◽  
Vol 12 (1) ◽  
pp. 118
Author(s):  
Meinershagen
Keyword(s):  
Hausdorff Measure

Derivation Bases and the Hausdorff Measure

1987 ◽  
Vol 13 (1) ◽  
pp. 223
Author(s):  
Meinershagen
Keyword(s):  
Hausdorff Measure

scholarly journals Fractional Order of Evolution Inclusion Coupled with a Time and State Dependent Maximal Monotone Operator

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1395
Author(s):  
Charles Castaing ◽  
Christiane Godet-Thobie ◽  
Le Xuan Truong
Keyword(s):  
Fractional Order ◽  
Monotone Operator ◽  
Maximal Monotone Operator ◽  
Separable Hilbert Space ◽  
Maximal Monotone ◽  
Fractional Flow ◽  
Order Problem ◽  
Absolutely Continuous ◽  
State Dependent ◽  
Fractional Differential

This paper is devoted to the study of evolution problems involving fractional flow and time and state dependent maximal monotone operator which is absolutely continuous in variation with respect to the Vladimirov’s pseudo distance. In a first part, we solve a second order problem and give an application to sweeping process. In a second part, we study a class of fractional order problem driven by a time and state dependent maximal monotone operator with a Lipschitz perturbation in a separable Hilbert space. In the last part, we establish a Filippov theorem and a relaxation variant for fractional differential inclusion in a separable Banach space. In every part, some variants and applications are presented.

On Finite Part Integrals and Hadamard-Type Fractional Derivatives

2018 ◽  
Vol 13 (9) ◽  
Author(s):  
Li Ma ◽  
Changpin Li
Keyword(s):  
Fractional Derivative ◽  
Singular Integral ◽  
Fractional Derivatives ◽  
Continuous Functions ◽  
Finite Part ◽  
Absolutely Continuous Functions ◽  
Absolutely Continuous ◽  
Fundamental Properties ◽  
Logarithmic Series ◽  
The Right

This paper is devoted to investigating the relation between Hadamard-type fractional derivatives and finite part integrals in Hadamard sense; that is to say, the Hadamard-type fractional derivative of a given function can be expressed by the finite part integral of a strongly singular integral, which actually does not exist. Besides, our results also cover some fundamental properties on absolutely continuous functions, and the logarithmic series expansion formulas at the right end point of interval for functions in certain absolutely continuous spaces.

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