Solvable random walk model with memory and its relations with Markovian models of anomalous diffusion

TL;DR

This paper introduces an exactly solvable path-dependent random walk model with memory, revealing transitions between diffusive, subdiffusive, and logarithmic growth regimes, and relating it to anomalous diffusion and Lévy flights.

Contribution

It presents a novel solvable model with memory-dependent dynamics, connecting Markovian and non-Markovian anomalous diffusion behaviors.

Findings

Memory reduces diffusion coefficient in weakly non-Markovian regime.

Transition to subdiffusion occurs when mean backward jump time diverges.

Logarithmic MSD growth appears with very long-range memory.

Abstract

Motivated by studies on the recurrent properties of animal and human mobility, we introduce a path-dependent random walk model with long range memory for which not only the mean square displacement (MSD) can be obtained exactly in the asymptotic limit, but also the propagator. The model consists of a random walker on a lattice, which, at a constant rate, stochastically relocates at a site occupied at some earlier time. This time in the past is chosen randomly according to a memory kernel, whose temporal decay can be varied via an exponent parameter. In the weakly non-Markovian regime, memory reduces the diffusion coefficient from the bare value. When the mean backward jump in time diverges, the diffusion coefficient vanishes and a transition to an anomalous subdiffusive regime occurs. Paradoxically, at the transition, the process is an anti-correlated L\'evy flight. Although in the subdiffusive regime the model exhibits some features of the continuous time random walk with infinite mean waiting time, it belongs to another universality class. If memory is very long-ranged, a second transition takes place to a regime characterized by a logarithmic growth of the MSD with time. In this case the process is asymptotically Gaussian and effectively described as a scaled Brownian motion with a diffusion coefficient decaying as 1/t.