Oscillatory Fractional Brownian Motion and Hierarchical Random Walks
arXiv:1201.5084 · doi:10.1007/s10440-013-9798-3
Abstract
We introduce oscillatory analogues of fractional Brownian motion, sub-fractional Brownian motion and other related long range dependent Gaussian processes, we discuss their properties, and we show how they arise from particle systems with or without branching and with different types of initial conditions, where the individual particle motion is the so-called c-random walk on a hierarchical group. The oscillations are caused by the discrete and ultrametric structure of the hierarchical group, and they become slower as time tends to infinity and faster as time approaches zero. We also give other results to provide an overall picture of the behavior of this kind of systems, emphasizing the new phenomena that are caused by the ultrametric structure as compared with results for analogous models on Euclidean space.
References in corpus (10)
- Some extensions of fractional Brownian motion and sub-fractional Brownian motion related to particle systems
- 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 branching systems with initial inhomogeneous Poisson states and related superprocesses
- Occupation Time Fluctuations of Weakly Degenerate Branching Systems
- Particle systems with quasi-homogeneous initial states and their occupation time fluctuations
- Particle picture interpretation of some Gaussian processes related to fractional Brownian motion
- Number variance for hierarchical random walks and related fluctuations
- Oscillatory Fractional Brownian Motion and Hierarchical Random Walks
- Occupation time fluctuations of Poisson and equilibrium branching systems in critical and large dimensions