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

Abstract

Consider the finite measure-valued continuous super-Brownian motion X on R d corresponding to the log-Laplace equation (∂/∂t)u= 1 2 Δu+βu−u 2, 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 k d | 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 k d replaced by K d > k d (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 δ 0. In this case, its expectation grows exponentially as t→∞.