Let X be the branching particle diffusion corresponding to the operator Lu + β(u 2 − u) on D ⊆ R d (where β ≥ 0 and β ≡ 0). Let λc denote the generalized principal eigenvalue for the operator L + β on D and assume that it is finite. When λc > 0 and L+β−λc satisfies certain spectral theoretical conditions, we prove that the random measure exp{−λct}Xt converges almost surely in the vague topology as t tends to infinity. This result is motivated by a cluster of articles due to Asmussen and Hering dating from the mid-seventies as well as the more recent work concerning analogous results for superdiffusions of [ET, EW]. We extend significantly the results in [AH76, AH77] and include some key examples of the branching process literature. As far as the proofs are concerned, we appeal to modern techniques concerning martingales and 'spine' decompositions or 'immortal particle pictures'.
Let X be either the branching diffusion corresponding to the operator Lu + β(u 2 − u) on D ⊆ R d [where β(x) ≥ 0 and β ≡ 0 is bounded from above] or the superprocess corresponding to the operator Lu + βu − αu 2 on D ⊆ R d (with α > 0 and β is bounded from above but no restriction on its sign). Let λ c denote the generalized principal eigenvalue for the operator L + β on D. We prove the following dichotomy: either λ c ≤ 0 and X exhibits local extinction or λ c > 0 and there is exponential growth of mass on compacts of D with rate λ c . For superdiffusions, this completes the local extinction criterion obtained by Pinsky [Ann. Probab. 24 (1996) 237-267] and a recent result on the local growth of mass under a spectral assumption given by Engländer and Turaev [Ann. Probab. 30 (2002) 683-722]. The proofs in the above two papers are based on PDE techniques, however the proofs we offer are probabilistically conceptual. For the most part they are based on "spine" decompositions or "immortal particle representations" along with martingale convergence and the law of large numbers. Further they are generic in the sense that they work for both types of processes.
Pinsky (1996) [15] proved that the finite mass superdiffusion X corresponding to the semilinear operator Lu + βu − αu 2 exhibits local extinction if and only if λ c ≤ 0, where λ c := λ c (L+β) is the generalized principal eigenvalue of L + β on R d . For the case when λ c > 0, it has been shown in Engländer and Turaev (2000) [8] that in law the superdiffusion locally behaves like exp[tλ c ] times a non-negative nondegenerate random variable, provided that the operator L + β − λ c satisfies a certain spectral condition ('product-criticality'), and that α and µ = X 0 are 'not too large'.In this article we will prove that the convergence in law used in the formulation in [8] can actually be replaced by convergence in probability. Furthermore, instead of R d we will consider a general Euclidean domain D ⊆ R d . As far as the proof of our main theorem is concerned, the heavy analytic method of [8] is replaced by a different, simpler and more probabilistic one. We introduce a space-time weighted superprocess (H-transformed superprocess) and use it in the proof along with some elementary probabilistic arguments.MSC 2000 subject classifications. 60J60, 60J80 Key words and phrases. super-Brownian motion, superdiffusion, superprocess, Law of Large Numbers, H-transform, weighted superprocess, scaling limit, local extinction, local survival. S'agissant de la preuve du theorème principale la lourde methode analytique de [8] est remplacée par une approche probabiliste plus simple. Nous introduisons une renormalisation spatio-temporelle du superprocessus ('H-transformed superprocess') que nous utilisons dans la preuve combinée a des arguments probabilistesélémentaires.
Consider the σ -finite measure-valued diffusion corresponding to the evolution equation u t = Lu + β(x)u − f (x, u), whereand n is a smooth kernel satisfying an integrability condition. We assume that β, α ∈ C η (R d ) with η ∈ (0, 1], and α > 0. Under appropriate spectral theoretical assumptions we prove the existence of the random measure(with respect to the vague topology), where λ c is the generalized principal eigenvalue of L + β on R d and it is assumed to be finite and positive, completing a result of Pinsky on the expectation of the rescaled process. Moreover, we prove that this limiting random measure is a nonnegative nondegenerate random multiple of a deterministic measure related to the operator L + β.When β is bounded from above, X is finite measure-valued. In this case, under an additional assumption on L + β, we can actually prove the existence of the previous limit with respect to the weak topology.As a particular case, we show that if L corresponds to a positive recurrent diffusion Y and β is a positive constant, then lim t ↑∞ e −βt X t (dx) exists and equals a nonnegative nondegenerate random multiple of the invariant measure for Y .Taking L = 1 2 on R and replacing β by δ 0 (super-Brownian motion with a single point source), we prove a similar result with λ c replaced by 1 2 and with the deterministic measure e −|x| dx, giving an answer in the affirmative to a problem proposed by Engländer and Fleischmann [Stochastic Process. Appl. 88 (2000) 37-58].The proofs are based upon two new results on invariant curves of strongly continuous nonlinear semigroups.
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