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Sample path properties of the local time of multifractional Brownian motion. (English) Zbl 1138.60032

The paper is devoted to various properties of the multifractal Brownian motion (mBm) ${B}^{H}=\left({B}^{H\left(t\right)}\left(t\right)$, $t>0\right)$, which is the Gaussian process with the Hurst exponent depending on time. The representation of the mBm was introduced by J. Lévy-Vehel and R. F. Peltier [“Multifractional Brownian motion: Definitions and preliminary results”, Technical Report PR-2645, INRIA (1995)] by

$\begin{array}{c}{B}^{H\left(t\right)}\left(t\right)=\frac{1}{{\Gamma }\left(H\left(t\right)+1/2\right)}\left({\int }_{-\infty }^{0}\left[{\left(t-u\right)}^{H\left(t\right)-1/2}-{\left(-u\right)}^{H\left(t\right)-1/2}\right]W\left(du\right)\hfill \\ \hfill +{\int }_{0}^{t}{\left(t-u\right)}^{H\left(t\right)-1/2}W\left(du\right)\right),\phantom{\rule{1.em}{0ex}}t\ge 0,\end{array}$

where $H$ is some Hölder continuous function and $W$ is the standard Brownian motion on $\left(-\infty ,\infty \right)$. Independently, mBm was defined by A. Benassi, S. Jaffard and D. Roux [Rev. Mat. Iberoam. 13, No. 1, 19–90 (1997; Zbl 0880.60053)] by

$\stackrel{^}{B}{\phantom{\rule{0.166667em}{0ex}}}^{H\left(t\right)}\left(t\right)={\int }_{ℝ}\frac{{e}^{it\xi }-1}{{|\xi |}^{H\left(t\right)+1/2}}d\stackrel{^}{W}\left(d\xi \right)·$

First, the authors establish the local and uniform moduli of continuity for the local time of the mBm. In contrast to the classical result, in the case of mBm the moduli of continuity depends on the point at which the regularity is studied [see also M. Csörgö, Z.-Y. Lin and Q.-M. Shao, Stochastic Processes Appl. 58, No. 1, 1–21 (1995; Zbl 0834.60088) and N. Kono, Proc. Japan Acad., Ser. A 53, 84–87 (1977; Zbl 0437.60057)]. Secondly, the authors establish the law of iterated logarithm in the Chung’s form.

Further, the authors establish the asymptotic self-similarity for the local times of the mBm. Then this results is used to prove some local limit theorems for mBm.

##### MSC:
 60G17 Sample path properties 60G15 Gaussian processes 60J55 Local time, additive functionals