Extreme value statistics and arcsine laws for heterogeneous diffusion processes

Abstract

Heterogeneous diffusion with spatially changing diffusion coefficient arises in many experimental systems like protein dynamics in the cell cytoplasm, mobility of cajal bodies and confined hardsphere fluids. Here, we showcase a simple model of heterogeneous diffusion where the diffusion coefficient D(x) varies in power-law way, i.e. D(x) ∼ |x| −α with the exponent α > −1. This model exhibits anomalous scaling of the mean squared displacement (MSD) of the form ∼ t 2 2+α and weak ergodicity breaking in the sense that ensemble averaged and time averaged MSDs do not converge. In this paper, we look at the extreme value statistics of this model and derive, for all α, the exact probability distributions of the maximum spatial displacement M (t) and arg-maximum tm(t) (i.e. the time at which this maximum is reached) till duration t. In the second part of our paper, we analyze the statistical properties of the residence time tr(t) and the last-passage time t (t) and compute their distributions exactly for all values of α. Our study unravels that the heterogeneous version (α = 0) displays many rich and contrasting features compared to that of the standard Brownian motion (SBm). For example, while for SBm (α = 0), the distributions of tm(t), tr(t) and t (t) are all identical (á la "arcsine laws" due to Lévy), they turn out to be significantly different for non-zero α. Another interesting property of tr(t) is the existence of a critical α (which we denote by αc = −0.3182) such that the distribution exhibits a local maximum at tr = t/2 for α < αc whereas it has minima at tr = t/2 for α ≥ αc. The underlying reasoning for this difference hints at the very contrasting natures of the process for α ≥ αc and α < αc which we thoroughly examine in our paper. All our analytical results are backed by extensive numerical simulations.

First-passage Brownian functionals with stochastic resetting

“…For this class of functionals, Z(x) = θ(x − x ), where θ(x − x ) denotes the Heaviside function. In the past, the residence time has been studied in various different scenarios such as diffusion in confinement [17], or in a potential landscape [18], in heterogeneous diffusion processes [19], in Brownian excursion processes [20] and active models [21]. While these studies have mostly considered residence time for fixed time, we here focus on a situation where the residence time is estimated upto the first passage event.…”