Extinction properties of super-Brownian motions with additional spatially dependent mass production

Consider the finite measure-valued continuous super-Brownian motion X on Rd corresponding to the log-Laplace equation (∂/∂t)u=12Δu+βu−u2, where the coefficient β(x) for the additional mass production varies in space, is Hölder continuous, and bounded from above. We prove criteria for (finite time) extinction and local extinction of X in terms of β. There exists a threshold decay rate kd|x|−2 as |x|→∞ such that X does not become extinct if β is above this threshold, whereas it does below the threshold (where for this case β might have to be modified on a compact set). For local extinction one has the same criterion, but in dimensions d>6 with the constant kd replaced by Kd>kd (phase transition). h-transforms for measure-valued processes play an important role in the proofs. We also show that X does not exhibit local extinction in dimension 1 if β is no longer bounded from above and, in fact, degenerates to a single point source δ0. In this case, its expectation grows exponentially as t→∞.