A diffusion model of surface soil pollution based on planar finite-velocity stochastic motion with random lifetime

TL;DR

This paper develops a diffusion model for surface soil pollution using planar finite-velocity stochastic motion with random lifetime, providing explicit formulas and numerical analysis for different particle lifetime distributions.

Contribution

It introduces a novel diffusion model based on planar Markov random flight with explicit stationary density formulas for exponential and gamma lifetime distributions.

Findings

Explicit stationary density formulas for exponential lifetime case.

Numerical and graphical analysis of pollution behavior over time.

Extension remarks on asymmetric stochastic motion models.

Abstract

We present a diffusion model of surface soil pollution from a stationary source based on the symmetric stochastic motion at finite speed in the plane $\Bbb R^2$, also called the planar Markov random flight, whose lifetime is a random variable with given distribution. We consider a heavy-particle model, in which the lifetime is supposed to be an exponentially-distributed random variable, and obtain an explicit formula for the stationary probability density of the pollution process expressed in terms of McDonald functions with variable indices. We also study a light-particle model, in which the lifetime is a gamma-distributed random variable. In this case, the stationary probability density of the pollution process is given in the form of a definite integral calculated numerically, as well as in the form of a functional series composed of the hypergeometric functions with variable coefficients. These stationary densities are plotted in a figure and numerically calculated tables that demonstrate the behaviour of the pollution process on long time intervals. Some remarks on the pollution model based on asymmetric finite-velocity planar stochastic motion are also given.