One-dimensional and planar random motions with variable propagation speeds

TL;DR

This paper investigates various types of one-dimensional and planar random motions with variable speeds, providing new mathematical tools and explicit formulas for moments, covariance, and limiting behaviors of these stochastic processes.

Contribution

It introduces a new family of polynomials for moment calculations and analyzes the hydrodynamic limits of complex random motions with variable velocities.

Findings

Derived general expressions for moments of the process.

Established explicit covariance functions for time-dependent velocity motions.

Conjectured absorption behavior in the hydrodynamic limit.

Abstract

In this paper, we study univariate and planar random motions with variable propagation speeds. We first consider motions with space-varying velocity, which can be reduced to constant-velocity motions by means of suitable nonlinear transformations. We examine a special case of a motion which is confined within the unit interval. To provide a general expression of the moments of this process, we introduce a new family of polynomials which generalize the classical Euler polynomials. We then examine a planar extension of this process which moves along orthogonal directions. A process with velocity depending on the direction is also examined, and its mean conditional on the initial direction and the number of direction changes is given in terms of confluent hypergeometric functions. We conjecture that, in the hydrodynamic limit, this process is absorbed at a point in an arbitrarily small time. We finally study a motion with time-dependent velocity. We prove that this process can be represented as an integral with respect to a standard telegraph process, and we obtain its covariance function explicitly. Moreover, we show that this process behaves as a It\^{o} integral with respect to Brownian motion in the hydrodynamic limit.