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Quantifying the non-ergodicity of scaled Brownian motion

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Original languageEnglish
Article number375002
JournalJournal of Physics A: Mathematical and Theoretical
Issue number37
Publication statusPublished - 18 Sep 2015
Publication typeA1 Journal article-refereed


We examine the non-ergodic properties of scaled Brownian motion (SBM), a non-stationary stochastic process with a time dependent diffusivity of the form $D(t)\simeq {t}^{\alpha -1}$. We compute the ergodicity breaking parameter EB in the entire range of scaling exponents α, both analytically and via extensive computer simulations of the stochastic Langevin equation. We demonstrate that in the limit of long trajectory lengths T and short lag times Δ the EB parameter as function of the scaling exponent α has no divergence at α = 1/2 and present the asymptotes for EB in different limits. We generalize the analytical and simulations results for the time averaged and ergodic properties of SBM in the presence of ageing, that is, when the observation of the system starts only a finite time span after its initiation. The approach developed here for the calculation of the higher time averaged moments of the particle displacement can be applied to derive the ergodic properties of other stochastic processes such as fractional Brownian motion.


  • ageing, anomalous diffusion, scaled Brownian motion

Publication forum classification

Field of science, Statistics Finland