Superdiffusions with super-exponential growth: Construction, mass and spread

Superdiffusions corresponding to differential operators of the form Lu + βu − αu² with mass creation (potential) terms β(·) that are 'large functions' are studied. Our construction for superdiffusions with large mass creations works for the branching mechanism βu − αu¹+^γ , 0 < γ < 1, as well. Let D ⊆ R^d be a domain in R^d. When β is large, the generalized principal eigenvalue λ_c of L + β in D is typically infinite. Let {T_t, t ≥ 0} denote the Schrödinger semigroup of L + β in D with zero Dirichlet boundary condition. Under the mild assumption that there exists an 0 < h ∈ C²(D) so that T_th is finite-valued for all t ≥ 0, we show that there is a unique M_loc(D)-valued Markov process that satisfies a log-Laplace equation in terms of the minimal nonnegative solution to a semilinear initial value problem. Although for super-Brownian motion (SBM) this assumption requires β to be less than quadratic, the quadratic case will be treated as well. When λ_c = ∞, the usual machinery, including martingale methods and PDE as well as other similar techniques cease to work effectively, both for the construction and for the investigation of the large time behavior of superdiffusions. In this paper, we develop the following two new techniques for the study of the local/global growth of mass and for the spread of superdiffusions: • a generalization of the Fleischmann-Swart 'Poisson-coupling,' linking superprocesses with branching diffusions; • the introduction of a new concept: the 'p-generalized principal eigenvalue.' The precise growth rate for the total population of SBM with α(x) = β(x) = 1 + |x|^p for p ∈ [0, 2] is given in this paper.