Exact first-passage time distributions for three random diffusivity models

• Main Authors: Grebenkov, D. S, Sposini, V, Metzler, R, Oshanin, G, Seno, F
• Format: Journal Article
• Language: English
• Published: 11-07-2020
• Subjects: Physics - Biological Physics; Physics - Statistical Mechanics
• DOI: 10.48550/arxiv.2007.05765

J. Phys. A: Math. Theor. 54, 04LT01 (2021) We study the extremal properties of a stochastic process $x_t$ defined by a
Langevin equation $\dot{x}_t=\sqrt{2 D_0 V(B_t)}\,\xi_t$, where $\xi_t$ is a
Gaussian white noise with zero mean, $D_0$ is a constant scale factor, and
$V(B_t)$ is a stochastic "diffusivity" (noise strength), which itself is a
functional of independent Brownian motion $B_t$. We derive exact, compact
expressions for the probability density functions (PDFs) of the first passage
time (FPT) $t$ from a fixed location $x_0$ to the origin for three different
realisations of the stochastic diffusivity: a cut-off case $V(B_t)
=\Theta(B_t)$ (Model I), where $\Theta(x)$ is the Heaviside theta function; a
Geometric Brownian Motion $V(B_t)=\exp(B_t)$ (Model II); and a case with
$V(B_t)=B_t^2$ (Model III). We realise that, rather surprisingly, the FPT PDF
has exactly the L\'evy-Smirnov form (specific for standard Brownian motion) for
Model II, which concurrently exhibits a strongly anomalous diffusion. For
Models I and III either the left or right tails (or both) have a different
functional dependence on time as compared to the L\'evy-Smirnov density. In all
cases, the PDFs are broad such that already the first moment does not exist.
Similar results are obtained in three dimensions for the FPT PDF to an
absorbing spherical target.