Occupation-Time Fluctuations of an Age-Dependent Branching System Driven by Poisson Immigration

We study the occupation-time fluctuations of a critical age-dependent branching particle system with immigration in ${\mathbb{R}}^d$. Immigrants arrive according to a homogeneous Poisson random measure in space and time. Each particle moves independently according to a symmetric $\alpha$-stable process and, at the end of its lifetime, either dies or splits into two offspring with equal probability. The lifetime distribution is allowed to have either finite mean or a heavy tail of index $\gamma \in (0,1]$. We investigate the asymptotic behavior of the centered occupation-time process under a suitable space-time scaling. Assuming $\alpha <d<(1+\gamma )\alpha ,$ we prove that the rescaled occupation-time fluctuations weakly converge as processes with values in the space of tempered distributions to a centered Gaussian process with an explicitly identified covariance structure. The normalization and the covariance depend on both the stability index $\alpha$ and the tail exponent $\gamma$. The limiting process is self-similar, possesses long-range dependence, and is neither Markovian nor a semimartingale. In contrast with the corresponding age-dependent branching system without immigration, the contribution of the initial population vanishes in the limit, and the asymptotic fluctuations are entirely determined by the immigration mechanism. When $\gamma =1$, our results recover the covariance structure previously obtained for branching systems with immigration and finite-mean lifetimes. The proofs rely on the space-time random field approach, Fourier analytic techniques, and asymptotic properties of renewal functions associated with the lifetime distribution.