Anna Talarczyk — Page 2

arXiv2006

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 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.

math.PR↗

arXiv2005

Occupation time fluctuations of an infinite variance branching system in large dimensions

We prove limit theorems for rescaled occupation time fluctuations of a (d,alpha,beta)-branching particle system (particles moving in R^d according to a spherically symmetric alpha-stable Levy process, (1+beta)-branching, 0 alpha(1+beta)/beta. The fluctuation processes are continuous but their limits are stable processes with independent increments, which have jumps. The convergence is in the sense of finite-dimensional distributions, and also of space-time random fields (tightness does not hold in the usual Skorohod topology). The results are in sharp contrast with those for intermediate dimensions, alpha/beta < d < d(1+beta)/beta, where the limit process is continuous and has long range dependence (this case is studied by Bojdecki et al, 2005). The limit process is measure-valued for the critical dimension, and S'(R^d)-valued for large dimensions. We also raise some questions of interpretation of the different types of dimension-dependent results obtained in the present and previous papers in terms of properties of the particle system.

math.PR↗