@article{GrebenkovSposiniMetzleretal.2020,
  author    = {Grebenkov, Denis S. and Sposini, Vittoria and Metzler, Ralf and Oshanin, Gleb and Seno, Flavio},
  title     = {Exact distributions of the maximum and range of random diffusivity processes},
  series = {New Journal of Physics},
  volume    = {23},
  journal   = {New Journal of Physics},
  publisher = {Dt. Physikalische Ges.},
  address   = {Bad Honnef},
  issn      = {1367-2630},
  doi       = {10.1088/1367-2630/abd313},
  pages     = {23},
  year      = {2020},
  abstract  = {We study the extremal properties of a stochastic process xt defined by the Langevin equation ẋₜ =√2Dₜ ξₜ, in which ξt is a Gaussian white noise with zero mean and Dₜ is a stochastic'diffusivity', defined as a functional of independent Brownian motion Bₜ.We focus on threechoices for the random diffusivity Dₜ: cut-off Brownian motion, Dₜt ∼ Θ(Bₜ), where Θ(x) is the Heaviside step function; geometric Brownian motion, Dₜ ∼ exp(-Bₜ); and a superdiffusive process based on squared Brownian motion, Dₜ ∼ B²ₜ. For these cases we derive exact expressions for the probability density functions of the maximal positive displacement and of the range of the process xₜ on the time interval ₜ ∈ (0, T).We discuss the asymptotic behaviours of the associated probability density functions, compare these against the behaviour of the corresponding properties of standard Brownian motion with constant diffusivity (Dₜ = D0) and also analyse the typical behaviour of the probability density functions which is observed for a majority of realisations of the stochastic diffusivity process.},
  language  = {en}
}
@misc{GrebenkovSposiniMetzleretal.2020,
  author    = {Grebenkov, Denis S. and Sposini, Vittoria and Metzler, Ralf and Oshanin, Gleb and Seno, Flavio},
  title     = {Exact distributions of the maximum and range of random diffusivity processes},
  series = {Postprints der Universit{\"a}t Potsdam : Mathematisch-Naturwissenschaftliche Reihe},
  journal   = {Postprints der Universit{\"a}t Potsdam : Mathematisch-Naturwissenschaftliche Reihe},
  number    = {1142},
  issn      = {1866-8372},
  doi       = {10.25932/publishup-50397},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-503976},
  pages     = {24},
  year      = {2020},
  abstract  = {We study the extremal properties of a stochastic process xt defined by the Langevin equation ẋₜ =√2Dₜ ξₜ, in which ξt is a Gaussian white noise with zero mean and Dₜ is a stochastic'diffusivity', defined as a functional of independent Brownian motion Bₜ.We focus on threechoices for the random diffusivity Dₜ: cut-off Brownian motion, Dₜt ∼ Θ(Bₜ), where Θ(x) is the Heaviside step function; geometric Brownian motion, Dₜ ∼ exp(-Bₜ); and a superdiffusive process based on squared Brownian motion, Dₜ ∼ B²ₜ. For these cases we derive exact expressions for the probability density functions of the maximal positive displacement and of the range of the process xₜ on the time interval ₜ ∈ (0, T).We discuss the asymptotic behaviours of the associated probability density functions, compare these against the behaviour of the corresponding properties of standard Brownian motion with constant diffusivity (Dₜ = D0) and also analyse the typical behaviour of the probability density functions which is observed for a majority of realisations of the stochastic diffusivity process.},
  language  = {en}
}