Occupation time limits of inhomogeneous Poisson systems of independent particles

Abstract

We prove functional limits theorems for the occupation time process of a system of particles moving independently in $R^d$ according to a symmetric $α$-stable Lévy process, and starting off from an inhomogeneous Poisson point measure with intensity measure $μ(dx)=(1+|x|^γ)^{-1}dx,γ>0$, and other related measures. In contrast to the homogeneous case $(γ=0)$, the system is not in equilibrium and ultimately it vanishes, and there are more different types of occupation time limit processes depending on arrangements of the parameters $γ, d$ and $α$. The case $γ<d<α$ leads to an extension of fractional Brownian motion.