@article{SposiniChechkinMetzler2018,
  author    = {Sposini, Vittoria and Chechkin, Aleksei V. and Metzler, Ralf},
  title     = {First passage statistics for diffusing diffusivity},
  series = {Journal of physics : A, Mathematical and theoretical},
  volume    = {52},
  journal   = {Journal of physics : A, Mathematical and theoretical},
  number    = {4},
  publisher = {IOP Publ. Ltd.},
  address   = {Bristol},
  issn      = {1751-8113},
  doi       = {10.1088/1751-8121/aaf6ff},
  pages     = {11},
  year      = {2018},
  abstract  = {A rapidly increasing number of systems is identified in which the stochastic motion of tracer particles follows the Brownian law < r(2)(t)> similar or equal to Dt yet the distribution of particle displacements is strongly non-Gaussian. A central approach to describe this effect is the diffusing diffusivity (DD) model in which the diffusion coefficient itself is a stochastic quantity, mimicking heterogeneities of the environment encountered by the tracer particle on its path. We here quantify in terms of analytical and numerical approaches the first passage behaviour of the DD model. We observe significant modifications compared to Brownian-Gaussian diffusion, in particular that the DD model may have a faster first passage dynamics. Moreover we find a universal crossover point of the survival probability independent of the initial condition.},
  language  = {en}
}
@misc{ŚlęzakBurneckiMetzler2019,
  author    = {Ślęzak, Jakub and Burnecki, Krzysztof and Metzler, Ralf},
  title     = {Random coefficient autoregressive processes describe Brownian yet non-Gaussian diffusion in heterogeneous systems},
  series = {Postprints der Universit{\"a}t Potsdam Mathematisch-Naturwissenschaftliche Reihe},
  journal   = {Postprints der Universit{\"a}t Potsdam Mathematisch-Naturwissenschaftliche Reihe},
  number    = {765},
  issn      = {1866-8372},
  doi       = {10.25932/publishup-43792},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-437923},
  pages     = {18},
  year      = {2019},
  abstract  = {Many studies on biological and soft matter systems report the joint presence of a linear mean-squared displacement and a non-Gaussian probability density exhibiting, for instance, exponential or stretched-Gaussian tails. This phenomenon is ascribed to the heterogeneity of the medium and is captured by random parameter models such as 'superstatistics' or 'diffusing diffusivity'. Independently, scientists working in the area of time series analysis and statistics have studied a class of discrete-time processes with similar properties, namely, random coefficient autoregressive models. In this work we try to reconcile these two approaches and thus provide a bridge between physical stochastic processes and autoregressive models.Westart from the basic Langevin equation of motion with time-varying damping or diffusion coefficients and establish the link to random coefficient autoregressive processes. By exploring that link we gain access to efficient statistical methods which can help to identify data exhibiting Brownian yet non-Gaussian diffusion.},
  language  = {en}
}
@article{ŚlęzakBurneckiMetzler2019,
  author    = {Ślęzak, Jakub and Burnecki, Krzysztof and Metzler, Ralf},
  title     = {Random coefficient autoregressive processes describe Brownian yet non-Gaussian diffusion in heterogeneous systems},
  series = {New Journal of Physics},
  volume    = {21},
  journal   = {New Journal of Physics},
  publisher = {Deutsche Physikalische Gesellschaft ; IOP, Institute of Physics},
  address   = {Bad Honnef und London},
  issn      = {1367-2630},
  doi       = {10.1088/1367-2630/ab3366},
  pages     = {18},
  year      = {2019},
  abstract  = {Many studies on biological and soft matter systems report the joint presence of a linear mean-squared displacement and a non-Gaussian probability density exhibiting, for instance, exponential or stretched-Gaussian tails. This phenomenon is ascribed to the heterogeneity of the medium and is captured by random parameter models such as 'superstatistics' or 'diffusing diffusivity'. Independently, scientists working in the area of time series analysis and statistics have studied a class of discrete-time processes with similar properties, namely, random coefficient autoregressive models. In this work we try to reconcile these two approaches and thus provide a bridge between physical stochastic processes and autoregressive models.Westart from the basic Langevin equation of motion with time-varying damping or diffusion coefficients and establish the link to random coefficient autoregressive processes. By exploring that link we gain access to efficient statistical methods which can help to identify data exhibiting Brownian yet non-Gaussian diffusion.},
  language  = {en}
}
@misc{ŚlęzakMetzlerMagdziarz2018,
  author    = {Ślęzak, Jakub and Metzler, Ralf and Magdziarz, Marcin},
  title     = {Superstatistical generalised Langevin equation},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-409315},
  pages     = {25},
  year      = {2018},
  abstract  = {Recent advances in single particle tracking and supercomputing techniques demonstrate the emergence of normal or anomalous, viscoelastic diffusion in conjunction with non-Gaussian distributions in soft, biological, and active matter systems. We here formulate a stochastic model based on a generalised Langevin equation in which non-Gaussian shapes of the probability density function and normal or anomalous diffusion have a common origin, namely a random parametrisation of the stochastic force. We perform a detailed analysis demonstrating how various types of parameter distributions for the memory kernel result in exponential, power law, or power-log law tails of the memory functions. The studied system is also shown to exhibit a further unusual property: the velocity has a Gaussian one point probability density but non-Gaussian joint distributions. This behaviour is reflected in the relaxation from a Gaussian to a non-Gaussian distribution observed for the position variable. We show that our theoretical results are in excellent agreement with stochastic simulations.},
  language  = {en}
}
@article{ŚlęzakMetzlerMagdziarz2018,
  author    = {Ślęzak, Jakub and Metzler, Ralf and Magdziarz, Marcin},
  title     = {Superstatistical generalised Langevin equation},
  series = {New Journal of Physics},
  volume    = {20},
  journal   = {New Journal of Physics},
  number    = {023026},
  publisher = {Deutsche Physikalische Gesellschaft / Institute of Physics},
  address   = {Bad Honnef und London},
  issn      = {1367-2630},
  doi       = {10.1088/1367-2630/aaa3d4},
  pages     = {1 -- 25},
  year      = {2018},
  abstract  = {Recent advances in single particle tracking and supercomputing techniques demonstrate the emergence of normal or anomalous, viscoelastic diffusion in conjunction with non-Gaussian distributions in soft, biological, and active matter systems. We here formulate a stochastic model based on a generalised Langevin equation in which non-Gaussian shapes of the probability density function and normal or anomalous diffusion have a common origin, namely a random parametrisation of the stochastic force. We perform a detailed analysis demonstrating how various types of parameter distributions for the memory kernel result in exponential, power law, or power-log law tails of the memory functions. The studied system is also shown to exhibit a further unusual property: the velocity has a Gaussian one point probability density but non-Gaussian joint distributions. This behaviour is reflected in the relaxation from a Gaussian to a non-Gaussian distribution observed for the position variable. We show that our theoretical results are in excellent agreement with stochastic simulations.},
  language  = {en}
}