Year: 2026
Type: preprint
Abstract: Suppose $X=(X_t)_{t\ge0}$ is a supercritical superprocess with a non-local branching mechanism on a Lusin space $E$, whose mean semigroup admits a positive leading eigenvalue $λ_1$ with a bounded positive eigenfunction $φ$ and a finite eigenmeasure $\widetildeφ$. Let $Y_t:=\int_0^t X_s\mathrm{d}s$ be its occupation measure. We first prove a law of large numbers, that is, for every bounded measurable function $f$, $e^{-λ_1 t}\langle f,Y_t\rangle$ converges in $L^2$ to $λ_1^{-1}\langle f,\widetildeφ\rangle W^φ_\infty$, where $W^φ_\infty$ is the limit of the martingale $W^φ_t:=e^{-λ_1 t}\langleφ,X_t\rangle$. We then establish functional central limit theorems for the fluctuations of $\langle f,Y_t\rangle$ around this random centering. We show that the normalisation and the limit are determined by the spectral decay rate of $f$ compared with the critical value $λ_1/2$, and there are three possible regimes: after suitable normalisation, the fluctuation process converges in distribution in $\mathcal{C}[0,+\infty)$ either to a mixed Gaussian process $\sqrt{W^φ_{\infty}}G(f)$, or to a constant Gaussian process with random variance depending on $W^φ_{\infty}$, or to a non-Gaussian limit given by the limit of a martingale associated with the spectral projection of $f$. These results are the occupation-time counterparts of the fluctuation theorems for instantaneous linear functionals of supercritical non-local branching superprocesses.
Source: arXiv (Cornell University)
Author: Yang, Ting
Institution: Beijing Institute of Technology
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7221548098