Exact distributions of the maximum and range of random diffusivity processes

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 $\xi_t$ 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, $D_t \sim \Theta(B_t)$, where $\Theta(x)$ is the Heaviside step function; geometric Brownian motion, $D_t \sim exp(−B_t)$; 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 $t \in (0, T)$. 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