Exact distributions of the maximum and range of random diffusivity processes

Abstract

We study the extremal properties of a stochastic process x _t defined by the Langevin equation ${\dot {x}}_{t}=\sqrt{2{D}_{t}}\enspace {\xi }_{t}$ , 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, ${D}_{t}\sim {B}_{t}^{2}$ . 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.

Keywords

• random diffusivity
• extremal values
• maximum and range
• diffusion, Brownian motion