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Local extinction versus local exponential growth for spatial branching processes

Let X be either the branching diffusion corresponding to the operator Lu+β(u2−u) on D⊆ Rd [where β(x)≥0 and β≡0 is bounded from above] or the superprocess corresponding to the operator Lu+βu−αu2 on D⊆ Rd (with α>0 and β is bounded from above but no restriction on its sign). Let λc denote the generalized principal eigenvalue for the operator L+β on D. We prove the following dichotomy: either λc≤0 and X exhibits local extinction or λc>0 and there is exponential growth of mass on compacts of D with rate λc. 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 Englander 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.