Nowhere differentiable functions constructed from probabilistic point of view
type: Item
instance of: scholarly article
title: Nowhere differentiable functions constructed from probabilistic point of view
zbMATH Open document ID: 0804.26004
DOI: 10.1007/BF01875156
published in: Acta Mathematica Hungarica
publication date: 1995-01-09T00:00:00Z
review text: The local time \(\alpha_ f(x,t)\) at \(x\) is a density function associated with a given function \(f\), mainly with a sample path of stochastic processes. The author uses the so-called Berman's principle to construct nowhere differentiable functions: the irregularity (regularity) of the original function \(f\) reflects regularity (irregularity) of the local time \(\alpha_ f(x,t)\). The author starts with the coordinate functions \((x(t),y(t))\) of the Peano curve. He regards them as stochastically independent random variables. Put \[ y_ 1(t)= x(t),\;y_ 2(t)= x(y(t)),\dots,y_ n(t)= x(y^{(n-1)}) \] and transform it to \(z_ n(t)= y_ n(t)- {1\over 2}\) to get, for \(\{a_ n\}\in l^ 2\), the function \[ f(t)= \sum^ \infty_{n=1} a_ n z_ n(t). \] Under certain circumstances \(f\) is nowhere differentiable, for example \[ f(t)= a\bigl(x(t)-\textstyle{{1\over 2}}\bigr)+b\bigl(x(y(t))- \textstyle{{1\over 2}}\bigr) \] with \(a,b\neq 0\), moreover, the level sets \(L_ x\) for all \(| x|< (| a|+ | b|)/2\) have Hausdorff dimension \({1\over 2}\).
reviewed by: Hermann Haase
Mathematics Subject Classification ID: 26A27
Mathematics Subject Classification ID: 60J55
zbMATH DE Number: 638257
zbMATH Keywords: local time
zbMATH Keywords: Berman's principle
zbMATH Keywords: nowhere differentiable functions
zbMATH Keywords: Peano curve
zbMATH Keywords: level sets
zbMATH Keywords: Hausdorff dimension
author: Norio Kôno
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 638257