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

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.

10 pages, 4 figures

References in corpus (5)

• Amnestically induced persistence in random walks
• Current fluctuations in stochastic systems with long-range memory
• Exact moments in a continuous time random walk with complete memory of its history
• Analytic Formulation, Exact Solutions, and Generalizations of the Elephant and the Alzheimer Random Walks
• Random Walkers with Shrinking Steps in d-Dimensions and Their Long Term Memory