*Tid:* **14 november 2011 kl 15.15-16.00.** (OBS! Dagen)
**Seminarierummet 3721**, Institutionen för
matematik, KTH, Lindstedts väg 25, plan 7.
Karta!
*Föredragshållare:*
**
Anna Talarczyk, University of Warsawa.
**
**Titel:**
Particle picture interpretation of some Gaussian processes
related to fractional Brownian motion.
**Abstract:**
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).
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