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Fractional Brownian density process and its self-intersection local time of order $k$. (English) Zbl 1074.60047
Summary: The fractional Brownian density process is a continuous centered Gaussian ${\cal S}'(\bbfR^d)$-valued process which arises as a high-density fluctuation limit of a Poisson system of independent $d$-dimensional fractional Brownian motions with Hurst parameter $H$. $({\cal S}'(\bbfR^d)$ is the space of tempered distributions.) The main result is that if the intensity measure $\mu$ of the (initial) Poisson random measure on $\bbfR^d$ is either the Lebesgue measure or a finite measure, then the density process has self-intersection local time of order $k\ge 2$ if and only if $Hd<k/(k-1)$. The latter is also a necessary and sufficient condition for the existence of multiple points of order $k$ for $d$-dimensional fractional Brownian motion, as proved by {\it M. Talagrand} [Probab. Theory Relat. Fields 112, 545--563 (1998; Zbl 0928.60026)]. This result extends to a non-Markovian case the relationship known for (Markovian) symmetric $\alpha$-stable Lévy processes and their corresponding density processes. New methods are used in order to overcome the lack of Markov property. Other properties of the fractional Brownian density process are also given, in particular the non-semimartingale property in the case $H\ne 1/2$, which is obtained by a general criterion for the non-semimartingale property of real Gaussian processes, is also proved.

60G15Gaussian processes
60G18Self-similar processes
60J55Local time, additive functionals
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