On a random walk with memory and its relation to Markovian processes

TL;DR

This paper investigates a one-dimensional random walk with memory, where step lengths adapt to minimize wandering, resulting in a non-diffusive process that can be described by an equivalent Markovian model.

Contribution

It introduces a novel memory-based random walk model with feedback that leads to non-diffusive behavior and establishes its equivalence to a Markovian process.

Findings

The walk exhibits non-diffusive behavior due to adaptive step lengths.

The position probability density matches that of a shrinking step random walk.

Two-time correlation functions differ despite similar position distributions.

Abstract

We study a one-dimensional random walk with memory in which the step lengths to the left and to the right evolve at each step in order to reduce the wandering of the walker. The feedback is quite efficient and lead to a non-diffusive walk. The time evolution of the displacement is given by an equivalent Markovian dynamical process. The probability density for the position of the walker is the same at any time as for a random walk with shrinking steps, although the two-time correlation functions are quite different.