Law of large numbers for superdiffusions: the non-ergodic case

In a previous paper of Winter and the author the Law of Large Numbers for the local mass of certain superdiffusions was proved under a spectral theoretical assumption, which is equivalent to the ergodicity (positive recurrence) of the motion component of an $H$-transformed (or weighted) superprocess. In fact the assumption is also equivalent to the property that the scaling for the expectation of the local mass is pure exponential. In this paper we go beyond ergodicity, that is we consider cases when the scaling is not purely exponential. Inter alia, we prove the analog of the Watanabe-Biggins Law of Large Numbers for super-Brownian motion (SBM). We will also prove another Law of Large Numbers for a bounded set moving with subcritical speed, provided the variance term decays sufficiently fast.