Spatial central limit theorem for supercritical superprocesses
A measure-valued diffusion (a superprocess) \(X_t\) can be seen as the short lifetime and high-density limit of a branching particle system. It is described by a couple \((L,\psi)\), where \(L\) is the infinitesimal generator of a strongly recurrent diffusion on \(\mathbb{R}^d\) and \(\Psi\) is a branching mechanism. If \(\varphi\) denotes the unique invariant measure of the diffusion generated by \(L\), a spatial law of large numbers for \(X_t\) has been proved in [\textit{J. Engländer} and \textit{A. Winter}, Ann. Inst. Henri Poincaré, Probab. Stat. 42, No. 2, 171--185 (2006; Zbl 1093.60058)], namely \[ \lim_{t \to \infty} e^{-\alpha t} \langle X_t, f\rangle = \langle \varphi, f\rangle V_\infty \] for any bounded continuous function \(f\). Here \(\alpha\) is the asymptotic growth rate of the system and \(V_\infty:=\lim_{t \to \infty} e^{-\alpha t} |X_t|\) is a non-trivial random variable. Here \(|X_t|=\langle X_t, 1\rangle\). The paper at hand provides a corresponding central limit theorem, i.e., convergence in distribution of \[ \frac{\langle X_t , f \rangle - |X_t| \langle \varphi, f \rangle}{N_t}. \] Three qualitatively different regimes (with different normalizations \(N_t\)) occur, depending on whether \(\alpha < 2 \mu\); \(\alpha=2 \mu\) or \(\alpha > 2 \mu\), where \(\mu\) is the exponential rate of convergence of a diffusion generated by \(L\) towards stationarity. This corresponds to the interplay of the growth governed by \(\psi\) and the spatial evolution governed by \(L\). In the first two regimes, Gaussian limits occur; while the limit in the third regime is conjectured to be non-Gaussian. Also, joint convergence of the spatial fluctuations and fluctuations of the total mass is provided, showing that they become asymptotically independent.
scientific article; zbMATH DE number 6862616
scientific article