Random Motions at Finite Speed in Higher Dimensions

Abstract

We present a general method of studying the transport process $\bold X(t)$ , t≥0, in the Euclidean space ℝ^m, m≥2, based on the analysis of the integral transforms of its distributions. We show that the joint characteristic functions of $\bold X(t)$ are connected with each other by a convolution-type recurrent relation. This enables us to prove that the characteristic function (Fourier transform) of $\bold X(t)$ in any dimension m≥2 satisfies a convolution-type Volterra integral equation of second kind. We give its solution and obtain the characteristic function of $\bold X(t)$ in terms of the multiple convolutions of the kernel of the equation with itself. An explicit form of the Laplace transform of the characteristic function in any dimension is given. The complete solution of the problem of finding the initial conditions for the governing partial differential equations, is given.

We also show that, under the standard Kac condition on the speed of the motion and on the intensity of the switching Poisson process, the transition density of the isotropic transport process converges to the transition density of the m-dimensional homogeneous Brownian motion with zero drift and diffusion coefficient depending on the dimension m.

We give the conditional characteristic functions of the isotropic transport process in terms of the inverse Laplace transform of the powers of the Gauss hypergeometric function. Some important models of the isotropic transport processes in lower dimensions are considered and some known results are derived as the particular cases of our general model by means of the method developed.