Supercritical Superprocesses: Proper Normalization and Non-degenerate Strong Limit

Suppose that $X=\{X_t, t\ge 0; \mathbb{P}_{\mu}\}$ is a supercritical superprocess in a locally compact separable metric space $E$. Let $\phi_0$ be a positive eigenfunction corresponding to the first eigenvalue $\lambda_0$ of the generator of the mean semigroup of $X$. Then $M_t:=e^{-\lambda_0t}\langle\phi_0, X_t\rangle$ is a positive martingale. Let $M_\infty$ be the limit of $M_t$. It is known (see, J. Appl. Probab. 46 (2009), 479--496) that $M_\infty$ is non-degenerate iff the $L\log L$ condition is satisfied. In this paper we are mainly interested in the case when the $L\log L$ condition is not satisfied. We prove that, under some conditions, there exist function $\gamma_t$ on $[0, \infty)$ and a non-degenerate random variable $W$ such that for any finite nonzero Borel measure $\mu$ on $E$, $$ \lim_{t\to\infty}\gamma_t\langle \phi_0,X_t\rangle =W,\qquad\mbox{a.s.-}\mathbb{P}_{\mu}. $$ We also give the almost sure limit of $\gamma_t\langle f,X_t\rangle$ for a class of general test functions $f$.