Supercritical superprocesses: Proper normalization and non-degenerate strong limit

Suppose that X = X t , t ⩾ 0;ℙ μ is a supercritical superprocess in a locally compact separable metric space E . Let φ 0 be a positive eigenfunction corresponding to the first eigenvalue λ 0 of the generator of the mean semigroup of X . Then M_t : = e^ - λ_0t⟨ϕ _0, X_t⟩ is a positive martingale. Let M ∞ be the limit of M t . It is known (see Liu et al. (2009)) that M ∞ is non-degenerate if and only if 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 a positive function γ t on [0, ∞) and a non-degenerate random variable W such that for any finite nonzero Borel measure μ on E , lim_t →∞ γ _t⟨ϕ _0, X_t⟩ = W, a.s. - ℙ_μ. We also give the almost sure limit of γ t 〈 f, X t 〉 for a class of general test functions f .