Anomalies in multifractal formalism for local time of Brownian motion (Q1284892)

By definition a measure \(\mu\) on \(I=[0,1]\) has the multifractal property if subsets \(I_\alpha\subset I\) having identical local dimension \(\alpha \) are fractals. In this case the multifractal spectrum of \(\mu\), i.e. \(\dim I_\alpha= f(\alpha)\), can be found with the help of multifractal formalism and box counting arguments using relations \(f(\alpha)= \tau^*(\alpha)\), \(f^*(q)= \tau (q)\), where * is the Legendre transform operation, while \(\tau= \tau_B (q)= \lim_{N\to \infty} \log\sum \mu^q (\Delta^N_i)/ \log\Delta_N\), \(| q | <\infty\), is a Rényi function (box \(\tau\)-function), \(\gamma= \{\Delta^N_i \}\) is a partition of \(I\) consisting of equal intervals of size \(\Delta_N=1/N\) and summation involves terms with \(\mu(\Delta_i)\neq 0\). An alternative definition of \(\tau\), analogous to Hausdorff dimension, is based on a changepoint \(\tau_H\) for \[ \Phi (q,\tau) =\sup_{\delta> 0}\inf_{\gamma (\delta)} \sum \mu^q (\Delta_i)/ | \Delta_i |^\tau, \] where \(\gamma(\delta)= \{\Delta_i :|\Delta_i |< \delta\}\) is the cover of the support of \(\mu\). The author studies Rényi functions for the local time measures \(L_h(dt)\) of fractional Brownian motion with an arbitrary self-similar index \(h\in(0,1)\) and discovers that for a Brownian motion \((h=1/2)\) \(\tau_B\neq\tau_H\). This fact should be of interest in physical applications, particularly, in turbulence theory.