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2007 Some Extensions of Fractional Brownian Motion and Sub-Fractional Brownian Motion Related to Particle Systems
Tomasz Bojdecki, Luis Gorostiza, Anna Talarczyk
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Electron. Commun. Probab. 12: 161-172 (2007). DOI: 10.1214/ECP.v12-1272

Abstract

In this paper we study three self-similar, long-range dependence, Gaussian processes. The first one, with covariance $$ \int^{s\wedge t}_0 u^a [(t-u)^b+(s-u)^b]du, $$ parameters $a>-1$, $-1 < b\leq 1$, $|b|\leq 1+a$, corresponds to fractional Brownian motion for $a=0$, $-1 < b < 1$. The second one, with covariance $$ (2-h)\biggl(s^h+t^h-\frac{1}{2}[(s+t)^h +|s-t|^h]\biggr), $$ parameter $0 < h\leq 4$, corresponds to sub-fractional Brownian motion for $0 < h < 2 $. The third one, with covariance $$ -\left(s^2\log s + t^2\log t -\frac{1}{2}[(s+t)^2 \log (s+t) +(s-t)^2 \log |s-t|]\right), $$ is related to the second one. These processes come from occupation time fluctuations of certain particle systems for some values of the parameters.

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Tomasz Bojdecki. Luis Gorostiza. Anna Talarczyk. "Some Extensions of Fractional Brownian Motion and Sub-Fractional Brownian Motion Related to Particle Systems." Electron. Commun. Probab. 12 161 - 172, 2007. https://doi.org/10.1214/ECP.v12-1272

Information

Accepted: 16 May 2007; Published: 2007
First available in Project Euclid: 6 June 2016

zbMATH: 1128.60025
MathSciNet: MR2318163
Digital Object Identifier: 10.1214/ECP.v12-1272

Subjects:
Primary: 60G18
Secondary: 60J80

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