Slow diffusion by Markov random flights

TL;DR

This paper introduces a model for slow diffusion in Euclidean spaces using Markov random flights driven by a Poisson process, detailing conditions for stationarity and presenting stationary distributions in low dimensions.

Contribution

It proposes a novel conception of slow diffusion processes based on random flights with small speeds and rates, providing stationary distributions in low-dimensional Euclidean spaces.

Findings

Conditions for slow diffusion leading to stationarity are established.

Stationary distributions for low-dimensional Euclidean spaces are derived.

Theoretical framework connecting random flights and diffusion processes is developed.

Abstract

We present a conception of the slow diffusion processes in the Euclidean spaces $\Bbb R^m, \; m\ge 1$, based on the theory of random flights with small constant speed that are driven by a homogeneous Poisson process of small rate. The slow diffusion conditions that, on long time intervals, lead to the stationary distributions, are given. The stationary distributions of slow diffusion processes in some Euclidean spaces of low dimensions, are presented.