Generalized arcsine laws for a sluggish random walker with subdiffusive growth

TL;DR

This work generalizes Lévy's arcsine laws to a sluggish one-dimensional random walker with a space-dependent diffusion constant $D(x)=|x|^{-\alpha}$ and drift $U(x)=|x|^{-\alpha}$, yielding subdiffusive late-time scaling $x(t)\sim t^{\mu}$ with $\mu=1/(\alpha+2)$. The authors derive exact scaling functions for the three classical observables—occupation time $t_+$, last passage time $t_{\rm l}$, and time of the maximum $t_M$—showing that while all reduce to arcsine at $\alpha=0$, they differ for any $\alpha>0$ and depend nontrivially on $\alpha$ via $\mu$. The derivations employ backward Feynman-Kac equations, Stieltjes transforms, and an $\epsilon$-path decomposition to obtain closed-form expressions and asymptotics for $f_+(\xi)$, $f_{\rm l}(\xi)$, and $f_M(\xi)$, with numerical simulations confirming the results. This work provides a rare example of a non-Brownian process where all three key observable distributions can be computed exactly, shedding light on subdiffusive transport and extreme-value statistics in inhomogeneous media.

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\geq 0 parametrizes 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\sim t^μ at late times with μ= 1/(α+2)\leq 1/2. In this paper, we compute exactly, for all α\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 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\geq 0. Numerical simulations are in excellent agreement with our analytical predictions.

Figures (8)

• Figure 1: Sample path of a stochastic process observed in a window $[0,T]$ together with the three observables considered in this paper: $t_+$ in \ref{['eq:t_plus_def']} is the sum of intervals on which the process is positive (red), $t_{\rm l}$ in \ref{['eq:last_passage_time_def']} (orange) and $t_M$ in \ref{['eq:time_of_max_def']} (purple). For completeness we also highlight the maximum value $M=x(t_M)$ reached by the process at time $t_M$ (purple). For Brownian motion the distributions of all three observables $t_+$, $t_{\rm l}$ and $t_M$ follow Lévy's arcsine law \ref{['eq:arcsine_law']}.
• Figure 2: A sluggish random walker hopping on a one dimensional lattice (with lattice constant $a=1$) according to the dynamical rules \ref{['eq:sluggish_rules']}. From a site at $x$, the walker moves to the left neighbour $x-1$ with probability $1/(|x|^{\alpha}+2)$, to the right neighbour $x+1$ with probability $1/(|x|^{\alpha}+2)$ and stays at site $x$ with the complementary probability $|x|^{\alpha}/(|x|^{\alpha}+2)$. The dashed curve represents the repulsive potential $U(x)= 1/|x|^{\alpha}$ felt by the walker as given in \ref{['du.1']}.
• Figure 3: (Solid lines) Probability distribution $f_+(\xi)$ computed analytically in \ref{['eq:f_plus']} for different values of $\alpha$. (Points) Filtered numerical samples of the occupation time probability $t_+ = \int_0^T \, \dd \tau \, \Theta(x_\tau)$. We have simulated $10^6$ runs of the walker and took an observation time of $T=10^6$.
• Figure 4: (Solid lines) Probability distribution $f_{\rm l}(\xi)$ computed analytically in \ref{['eq:f_l']} for different values of $\alpha$. (Points) Numerical obtained scaling function $f_{\rm l}(\xi)$. Simulation details are same as in Fig. \ref{['fig:f_plus']}.
• Figure 5: (Solid lines) Probability distribution $f_{\rm M}(\xi)$ computed analytically in \ref{['eq:f_M']} for different values of $\alpha$. (Points) Numerically obtained scaling function $f_M(\xi)$. Simulation details are same as in Fig. \ref{['fig:f_plus']}.