Random walks in non homogeneous Poissonian environment

TL;DR

This paper studies the asymptotic behavior of a particle moving in a non-homogeneous Poissonian environment, establishing conditions for weak convergence of normalized paths and constructing diffusion approximations for the process.

Contribution

It introduces new conditions for weak convergence of the particle's path and develops diffusion approximations using the parametric method in a non-homogeneous setting.

Findings

Weak convergence of normalized particle paths under certain conditions.

Diffusion approximations for the Markov chain model of the process.

Analysis of the accuracy of the diffusion approximations.

Abstract

We consider the moving particle process in Rd which is defined in the following way. There are two independent sequences (Tk) and (dk) of random variables. The variables Tk are non negative and form an increasing sequence, while variables dk form an i.i.d sequence with common distribution concentrated on the unit sphere. The values dk are interpreted as the directions, and Tk as the moments of change of directions. A particle starts from zero and moves in the direction d1 up to the moment T1 . It then changes direction to d2 and moves on within the time interval T2 minus T1 , etc. The speed is constant at all sites. The position of the particle at time t is denoted by X(t). We suppose that the points (Tk) form a non homogeneous Poisson point process and we are interested in the global behavior of the process (X(t)), namely, we are looking for conditions under which the processes (Y(T,t), T is non negative), Y(T,t) is X(tT) normalized by B(T), t in (0, 1), weakly converges in C(0, 1) to some process Y when T tends to infinity. In the second part of the paper the process X(t) is considered as a Markov chain. We construct diffusion approximations for this process and investigate their accuracy. The main tool in this part is the paramertix method.