Spines, skeletons and the strong law of large numbers for superdiffusions
The paper investigates almost sure limit behavior of a supercritical superdiffusion \({({X_t})_{t \geqslant 0}}\) on a domain \(D \subseteq {{\mathbb R}^d}\) with branching mechanism \[ (x,z) \mapsto - \beta (x) + \alpha (x){z^2} + \int_{(0,\infty )} {({e^{ - zy}} - 1 + zy)\Pi (x,dy)}. \] It is shown that for suitable test functions \(f\) and starting measures \(\mu \), \[ \frac{{\left\langle {f,{X_t}} \right\rangle }}{{{P_\mu }\left[ {\left\langle {f,{X_t}} \right\rangle } \right]}} \to {W_\infty } \] \({P_\mu }\)-almost surely as \(t \to \infty \), where \({W_\infty }\) is a finite, non-deterministic random variable characterized as a martingale limit. The authors use the skeleton decomposition that represents the superprocess as an immigration process along a branching particle process (the skeleton), where immigration occurs in a Poisson way along the space-time trajectories and at the branch points of the skeleton.
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