Probability Density of a Moving Particle

Abstract

This paper studies a particular random tour (continuous random walk): A particle moves in two dimensions at constant speed by choosing successive travel directions that are independent and uniformly distributed between 0 and 2π, with the lengths of the steps between direction changes being independent, exponentially distributed, random variables. An analytic expression for the probability density of the particle's position after a time t is derived, and an application is made to a military situation where the “particle” is a target trying to escape being destroyed by an unseen enemy. The bulk of the paper is devoted to deriving the density function.