Nowhere differentiable functions constructed from probabilistic point of view
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 \({1\over 2}\).
scientific article; zbMATH DE number 638257