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

Superdi usions corresponding to di erential operators of the form Lu+βu−αu with mass creation (potential) terms β(·) that are `large functions' are studied. Our construction for superdi usions with large mass creations works for the branching mechanism βu−αu , 0 < γ < 1, as well. Let D ⊆ R be a domain in R. When β is large, the generalized principal eigenvalue λc of L+β in D is typically in nite. Let {Tt, 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 ∈ C(D) so that Tth is nite-valued for all t ≥ 0, we show that there is a uniqueMloc(D)-valued Markov process that satis es 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 λc = ∞, the usual machinery, including martingale methods and PDE as well as other similar techniques cease to work e ectively, both for the construction and for the investigation of the large time behavior of superdi usions. 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 superdi usions: • a generalization of the Fleischmann-Swart `Poisson-coupling,' linking superprocesses with branching di usions; • 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| for p ∈ [0, 2] is given in this paper.