@misc{ŚlęzakMetzlerMagdziarz2019,
  author    = {Ślęzak, Jakub and Metzler, Ralf and Magdziarz, Marcin},
  title     = {Codifference can detect ergodicity breaking and non-Gaussianity},
  series = {Postprints der Universit{\"a}t Potsdam Mathematisch-Naturwissenschaftliche Reihe},
  journal   = {Postprints der Universit{\"a}t Potsdam Mathematisch-Naturwissenschaftliche Reihe},
  number    = {748},
  doi       = {10.25932/publishup-43617},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-436178},
  pages     = {25},
  year      = {2019},
  abstract  = {We show that the codifference is a useful tool in studying the ergodicity breaking and non-Gaussianity properties of stochastic time series. While the codifference is a measure of dependence that was previously studied mainly in the context of stable processes, we here extend its range of applicability to random-parameter and diffusing-diffusivity models which are important in contemporary physics, biology and financial engineering. We prove that the codifference detects forms of dependence and ergodicity breaking which are not visible from analysing the covariance and correlation functions. We also discuss a related measure of dispersion, which is a nonlinear analogue of the mean squared displacement.},
  language  = {en}
}
@article{ŚlęzakMetzlerMagdziarz2019,
  author    = {Ślęzak, Jakub and Metzler, Ralf and Magdziarz, Marcin},
  title     = {Codifference can detect ergodicity breaking and non-Gaussianity},
  series = {New Journal of Physics},
  volume    = {21},
  journal   = {New Journal of Physics},
  publisher = {Deutsche Physikalische Gesellschaft},
  address   = {Bad Honnef},
  issn      = {1367-2630},
  doi       = {10.1088/1367-2630/ab13f3},
  pages     = {25},
  year      = {2019},
  abstract  = {We show that the codifference is a useful tool in studying the ergodicity breaking and non-Gaussianity properties of stochastic time series. While the codifference is a measure of dependence that was previously studied mainly in the context of stable processes, we here extend its range of applicability to random-parameter and diffusing-diffusivity models which are important in contemporary physics, biology and financial engineering. We prove that the codifference detects forms of dependence and ergodicity breaking which are not visible from analysing the covariance and correlation functions. We also discuss a related measure of dispersion, which is a nonlinear analogue of the mean squared displacement.},
  language  = {en}
}
@misc{MolinaGarciaSandevSafdarietal.2019,
  author    = {Molina-Garcia, Daniel and Sandev, Trifce and Safdari, Hadiseh and Pagnini, Gianni and Chechkin, Aleksei V. and Metzler, Ralf},
  title     = {Crossover from anomalous to normal diffusion},
  series = {Postprints der Universit{\"a}t Potsdam Mathematisch-Naturwissenschaftliche Reihe},
  journal   = {Postprints der Universit{\"a}t Potsdam Mathematisch-Naturwissenschaftliche Reihe},
  number    = {507},
  issn      = {1866-8372},
  doi       = {10.25932/publishup-42259},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-422590},
  pages     = {28},
  year      = {2019},
  abstract  = {Abstract The emerging diffusive dynamics in many complex systems show a characteristic crossover behaviour from anomalous to normal diffusion which is otherwise fitted by two independent power-laws. A prominent example for a subdiffusive-diffusive crossover are viscoelastic systems such as lipid bilayer membranes, while superdiffusive-diffusive crossovers occur in systems of actively moving biological cells. We here consider the general dynamics of a stochastic particle driven by so-called tempered fractional Gaussian noise, that is noise with Gaussian amplitude and power-law correlations, which are cut off at some mesoscopic time scale. Concretely we consider such noise with built-in exponential or power-law tempering, driving an overdamped Langevin equation (fractional Brownian motion) and fractional Langevin equation motion. We derive explicit expressions for the mean squared displacement and correlation functions, including different shapes of the crossover behaviour depending on the concrete tempering, and discuss the physical meaning of the tempering. In the case of power-law tempering we also find a crossover behaviour from faster to slower superdiffusion and slower to faster subdiffusion. As a direct application of our model we demonstrate that the obtained dynamics quantitatively describes the subdiffusion-diffusion and subdiffusion-subdiffusion crossover in lipid bilayer systems. We also show that a model of tempered fractional Brownian motion recently proposed by Sabzikar and Meerschaert leads to physically very different behaviour with a seemingly paradoxical ballistic long time scaling.},
  language  = {en}
}