Occupation time limits of inhomogeneous Poisson systems of independent particles

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 from an inhomogeneous Poisson point measure with intensity measure μ ( d x ) = ( 1 + | x | γ ) − 1 d x , γ > 0 , and other related measures. In contrast to the homogeneous case ( γ = 0 ) , the system is not in equilibrium and ultimately it becomes locally extinct in probability, 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.