Area under subdiffusive random walks

We study the statistical properties of the area and the absolute area under the trajectories of subdiffusive random walks. Using different frameworks to describe subdiffusion (as the scaled Brownian motion, fractional Brownian motion, the continuous-time random walk or the Brownian motion in heterogeneous media), we compute the first two moments, the ergodicity breaking parameter for the absolute area and infer a general scaling for the probability density functions of these functionals. We discuss the differences between the statistical properties of the area and the absolute area for the different subdiffusion models and illustrate the experimental interest of our results. Our theoretical findings are supported by Monte Carlo simulations showing an excellent agreement.

💡 Research Summary

This paper investigates the statistical properties of two functionals of subdiffusive stochastic processes: the signed area A(t)=∫₀ᵗ x(τ)dτ under a trajectory and the absolute area |A(t)|=∫₀ᵗ |x(τ)|dτ. Four widely used models of anomalous diffusion are considered: Scaled Brownian Motion (SBM) with a time‑dependent diffusion coefficient, fractional Brownian motion (fBM) with long‑range correlations, a continuous‑time random walk (CTRW) with heavy‑tailed waiting times, and heterogeneous Brownian motion (HBM) where the diffusion coefficient varies as a power of the spatial coordinate.

The authors first present a general framework for stochastic functionals. For a functional Z(t)=∫₀ᵗ U