Fractional Brownian Density Process and Its Self-Intersection Local Time of Order k

Abstract

The fractional Brownian density process is a continuous centered Gaussian \(S\)′(ℝ 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. (\(S\)′(ℝ d ) is the space of tempered distributions). The main result proved in the paper is that if the intensity measure μ of the (initial) Poisson random measure on ℝ d is either the Lebesgue measure or a finite measure, then the density process has self-intersection local time of order k ≥ 2 if and only if Hd < k/(k − 1). The latter is also the necessary and sufficient condition for existence of multiple points of order k for d-dimensional fractional Brownian motion, as proved by Talagrand12. This result extends to a non-Markovian case the relationship known for (Markovian) symmetric α-stable Levy 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 ≠ 1/2, which is obtained by a general criterion for the non-semimartingale property of real Gaussian processes that we also prove.