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 Langevin equation x ̇ t = 2 D ξ , in which ξ is Gaussian white noise with zero mean and D ‘diffusivity’, as functional independent Brownian motion B . focus on three choices for random diffusivity : cut-off motion, ∼ Θ( ), where ) Heaviside step function; geometric exp(− ); superdiffusive based squared ${D}_{t}\sim {B}_{t}^{2}$?> ∼ B For these cases we derive exact expressions probability density functions maximal positive displacement range time interval ∈ (0, T ). discuss asymptotic behaviours associated functions, compare against behaviour corresponding standard constant ( = 0 also analyse typical observed majority realisations process.