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

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.