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

We apply limiting procedures to certain particle systems which give fractional Brownian motion (fBm), subfractional Brownian motion (sfBm), negative subfractional Brownian motion (nsfBm) and the odd part of fractional Brownian motion in the sense of Dzhaparidze and Van Zanten (2004). Fractional Brownian motion is a well known process with many applications. FBm with Hurst parameter H can be characterized as the only Gaussian process which has stationary increments and is self similar with index H. This is why fBm arises naturally in many situations. Subfractional Brownian motion and negative subfractional Brownian motion are Gaussian processes which are closely related to fBm. Previously, they only appeared for a narrow range of parameters in the context of occupation time fluctuations of branching particle systems. The odd part of fBm had not been given any physical interpretation at all.

Here we present several limit theorems related to particle systems with and without branching, which lead to fBm, sfBm, nsfBm and the odd part of fBm, and cover the full range of parameters. One of the approaches consists in representing fBm, sfBm and the odd part of fBm as <X(1),I_[0,t]>, <X(1), I_[0,t] - I_[-t, 0]> and <X(1), I_[-t,t]> respectively, where X(1) is an (extended) S'-valued random variable obtained as the fluctuation limit of either the empirical process or the occupation time process of an appropriate particle system.

The talk is based on joint work with Tomasz Bojdecki (arXiv:1108.2745v1) and earlier articles with Tomasz Bojdecki and Luis Gorostiza (see references in the first mentioned preprint).