Some extensions of fractional Brownian motion and sub-fractional Brownian motion related to particle systems

Abstract

In this paper we study three self-similar, long-range dependence, Gaussian processes. The first one, with covariance \int_0^{s\wedge t} 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)(s^h+t^h-[(s+t)^h +|s-t|^h]/2), parameter 0<h\leq 4, corresponds to sub-fractional Brownian motion for 0<h<2. The third one, with covariance -(s^2\log s + t^2\log t -[(s+t)^2 \log (s+t) +(s-t)^2 \log |s-t|]/2), is related to the second one. These processes come from occupation time fluctuations of certain particle systems for some values of the parameters.

11 pages; minor corrections and changes in notation

References in corpus (3)

• Occupation time limits of inhomogeneous Poisson systems of independent particles
• Self-similar stable processes arising from high-density limits of occupation times of particle systems
• Sample Path Properties of Bifractional Brownian Motion

Cited by in corpus (16)

• Occupation time limits of inhomogeneous Poisson systems of independent particles
• Self-similar stable processes arising from high-density limits of occupation times of particle systems
• Occupation times of subcritical branching immigration system with Markov motions
• A generalisation of the fractional Brownian field based on non-Euclidean norms
• Particle picture interpretation of some Gaussian processes related to fractional Brownian motion
• Bifractional Brownian motion for $H>1$ and $2HK\le 1$
• The Bouleau-Yor identity for a bi-fractional Brownian motion
• Small-time asymptotics for Gaussian self-similar stochastic volatility models
• Oscillatory Fractional Brownian Motion and Hierarchical Random Walks
• Bifractional Brownian motion: existence and border cases
• Fluctuations of the occupation times for branching system starting from infinitely divisible point processes
• Convergence to Weighted Fractional Brownian Sheets
• Occupation time fluctuation limits of infinite variance equilibrium branching systems
• Weak convergence towards two independent Gaussian processes from a unique Poisson process
• The quadratic covariation for a weighted fractional Brownian motion
• Self-similar Gaussian Markov processes