Generalized arcsine laws for a sluggish random walker with subdiffusive growth

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|^{-\alpha} and a drift potential U(x)=|x|^{-\alpha}, where \alpha\geq 0 parametrizes the model. For \alpha=0 it reduces to the standard diffusion, while for \alpha>0 it leads to a slow subdiffusive dynamics with the distance scaling as x\sim t^{\mu} at late times with \mu= 1/(\alpha+2)\leq 1/2. In this paper, we compute exactly, for all \alpha\ge 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_{\rm l} through the origin before T, and (iii) the time t_M at which the walker is maximally displaced on the positive side of the origin. We show that while for \alpha=0 all three distributions are identical and exhibit the celebrated arcsine laws of L\'evy, they become different from each other for any \alpha>0 and have nontrivial shapes dependent on \alpha. This generalizes the L\'evy's three arcsine laws for normal diffusion (\alpha=0) to the subdiffusive sluggish walker model with a general \alpha\geq 0. Numerical simulations are in excellent agreement with our analytical predictions.