Exact distributions of the maximum and range of random diffusivity processes

Abstract

Abstract We study the extremal properties of a stochastic process x t defined by the Langevin equation x ̇ t = 2 D t ξ t , 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, D t ∼ B t 2 . 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.