Let X be either the branching diffusion corresponding to the operator Lu + beta(u(2) - u) on D subset of or equal to R-d [where beta(x) greater than or equal to 0 and beta not equivalent to 0 is bounded from above] or the superprocess corresponding to the operator Lu + betau - alphau(2) on D subset of or equal to R-d (with alpha > 0 and beta is bounded from above but no restriction on its sign). Let lambda(c) denote the generalized principal eigenvalue for the operator L + beta on D. We prove the following dichotomy: either lambda(c) less than or equal to 0 and X exhibits local extinction or lambda(c) > 0 and there is exponential growth of mass on compacts of D with rate lambdac. 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) 683-7221. 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.

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An important property of Kingman's coalescent is that, starting from a state
with an infinite number of blocks, over any positive time horizon, it
transitions into an almost surely finite number of blocks. This is known as
`coming down from infinity'. Moreover, of the many different (exchangeable)
stochastic coalescent models, Kingman's coalescent is the `fastest' to come
down from infinity. In this article we study what happens when we counteract
this `fastest' coalescent with the action of an extreme form of fragmentation.
We augment Kingman's coalescent, where any two blocks merge at rate $c>0$, with
a fragmentation mechanism where each block fragments at constant rate,
$\lambda>0$, into it's constituent elements. We prove that there exists a phase
transition at $\lambda=c/2$, between regimes where the resulting `fast'
fragmentation-coalescence process is able to come down from infinity or not. In
the case that $\lambda