Abstract
We study the extremal properties of a stochastic process x t defined by the Langevin equation , in which ξ t is a Gaussian white noise with zero mean and D t is a stochastic ‘diffusivity’, defined as a functional of independent Brownian motion B t . We focus on three choices for the random diffusivity D t : cut-off Brownian motion, D t ∼ Θ(B t ), where Θ(x) is the Heaviside step function; geometric Brownian motion, D t ∼ exp(−B t ); and a superdiffusive process based on squared Brownian motion, . For these cases we derive exact expressions for the probability density functions of the maximal positive displacement and of the range of the process x t 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 (D t = D 0) and also analyse the typical behaviour of the probability density functions which is observed for a majority of realisations of the stochastic diffusivity process.
Highlights
The statistics of extreme values (EVs) of stochastic processes has been in the focus of extensive research in the mathematical and physical literature over several decades
We focus on three choices for the random diffusivity Dt: cut-off Brownian motion, Dt ∼ 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
Deviations from standard Brownian motion have been measured in a vast range of systems, starting with Richardson’s cubic law for the relative diffusion of tracers in turbulent media in 1926 [87]
Summary
Denis S Grebenkov1,2 , Vittoria Sposini2,3 , Ralf Metzler2,∗ , Gleb Oshanin and Flavio Seno. Cedex 05, France 5 INFN, Padova Section and Department of Physics and Astronomy ‘Galileo Galilei’, University of Padova, 35131 Padova, Italy ∗ Author to whom any correspondence should be addressed
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