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 $\alpha$-stable L\'evy process, and starting off from an inhomogeneous Poisson point measure with intensity measure $\mu(dx)=(1+|x|^{\gamma})^{-1}dx,\gamma>0$, and other related measures. In contrast to the homogeneous case $(\gamma=0)$, 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 $\gamma, d$ and $\alpha$. The case $\gamma<d<\alpha$ leads to an extension of fractional Brownian motion.