Random walk with heterogeneous sojourn time

TL;DR

This paper introduces a non-Markovian discrete-time random walk model with heterogeneous sojourn times on a 1D lattice, demonstrating its convergence to a diffusion equation and validating it through simulations.

Contribution

It develops a novel non-Markovian random walk model with variable sojourn times and proves its convergence to a continuum diffusion equation.

Findings

Discrete density converges to a diffusion equation solution

Green's function of the diffusion equation is derived

Monte Carlo simulations validate the model and theoretical results

Abstract

We introduce a discrete-time random walk model on a one-dimensional lattice with a nonconstant sojourn time and prove that the discrete density converges to a solution of a continuum diffusion equation. Our random walk model is not Markovian due to the heterogeneity in the sojourn time, in contrast to a random walk model with a nonconstant walk length. We derive a Markovian process by choosing appropriate subindexes of the time-space grid points, and then show the convergence of its discrete density through the parabolic-scale limit. We also find the Green's function of the continuum diffusion equation and present three Monte Carlo simulations to validate the random walk model and the diffusion equation.