Particle picture interpretation of some Gaussian processes related to fractional Brownian motion

TL;DR

This paper constructs and interprets various Gaussian processes related to fractional Brownian motion through particle systems, providing new physical insights and generalizations for a broad range of parameters.

Contribution

It introduces a particle system approach to represent and interpret Gaussian processes associated with fractional Brownian motion across full Hurst parameter ranges, including new processes and multidimensional fields.

Findings

Particle system representations of fBm, sub-fBm, nsfBm, and odd fBm parts.

Extension of interpretations to full Hurst parameter ranges.

Introduction of new Gaussian processes and multidimensional fields.

Abstract

We construct fractional Brownian motion (fBm), sub-fractional Brownian motion (sub-fBm), negative sub-fractional Brownian motion (nsfBm) and the odd part of fBm in the sense of Dzhaparidze and van Zanten (2004) by means of limiting procedures applied to some particle systems. These processes are obtained for full ranges of Hurst parameter. Particle picture interpretations of sub-fBm and nsfBm were known earlier (using a different approach) for narrow ranges of parameters; the odd part of fBm process had not been given any physical interpretation at all. Our approach consists in representing these processes as $<X(1),1_{[0,t]}>$, $<X(1),1_{[0,t]}-1_{[-t,0]}>$, $<X(1),1_{[-t,t]}>$, respectively, where X(1) is an (extended) $S'$-random variable obtained as the fluctuation limit of either empirical process or the occupation time process of an appropriate particle system. In fact, our construction is more general, permitting to obtain some new Gaussian processes, as well as multidimensional random fields. In particular, we generalize and presumably simplify some results by Hambly and Jones (2007). We also obtain a new class of $S'$-valued density processes, containing as a particular case the density process of Martin-L\"of (1976).