Year: 2025
Type: article
Abstract: We study a simple one dimensional sluggish random walk model with subdiffusive growth. In the continuum hydrodynamic limit, the model corresponds to a particle diffusing on a line with a space dependent diffusion constant D ( x ) = | x | − α and a drift potential U ( x ) = | x | − α , where α ⩾ 0 parameterizes the model. For α = 0 it reduces to the standard diffusion, while for α > 0 it leads to a slow subdiffusive dynamics with the distance scaling as x ∼ t μ at late times with μ = 1 / ( α + 2 ) ⩽ 1 / 2 . In this paper, we compute exactly, for all α ⩾ 0 , the full probability distributions of three observables for a sluggish walker of duration T starting at the origin: (i) the occupation time t + denoting the time spent on the positive side of the origin, (ii) the last passage time t l through the origin before T, and (iii) the time tM at which the walker is maximally displaced on the positive side of the origin. We show that while for α = 0 all three distributions are identical and exhibit the celebrated arcsine laws of Lévy, they become different from each other for any α > 0 and have nontrivial shapes dependent on α. This generalizes the Lévy’s three arcsine laws for normal diffusion ( α = 0 ) to the subdiffusive sluggish walker model with a general α ⩾ 0 . Numerical simulations are in excellent agreement with our analytical predictions.
Language: English
Subfield: Modeling and Simulation
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Funder: Agence Nationale de la Recherche
OpenAlex ID: W4407992678