Anomalous diffusion and generalized Sparre-Andersen scaling

TL;DR

This paper investigates the long-time behavior of anomalous diffusion processes, revealing non-Markovian characteristics through deviations from classical Sparre-Andersen scaling, and introduces a framework to distinguish memory effects.

Contribution

It provides a novel analysis of the survival probability deviations to identify non-Markovian features in anomalous diffusion, extending the understanding of scaling limits.

Findings

Anomalous diffusion can be non-Markovian despite classical scaling behaviors.

Survival probability deviations serve as indicators of memory effects.

The framework differentiates between truly memoryless and non-Markov processes.

Abstract

We are discussing long-time, scaling limit for the anomalous diffusion composed of the subordinated L\'evy-Wiener process. The limiting anomalous diffusion is in general non-Markov, even in the regime, where ensemble averages of a mean-square displacement or quantiles representing the group spread of the distribution follow the scaling characteristic for an ordinary stochastic diffusion. To discriminate between truly memory-less process and the non-Markov one, we are analyzing deviation of the survival probability from the (standard) Sparre-Andersen scaling.