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

Consider the finite measure-valued continuous super-Brownian motion X on R-d corresponding to the log-Laplace equation (partial derivative/partial derivative t)u = 1/2 Delta u + beta u - u(2), where the coefficient beta(x) for the additional mass production varies in space, is Holder continuous, and bounded from above. We prove criteria for (finite time) extinction and local extinction of X in terms of beta. There exists a threshold decay rate k(d)\ x \(-2) as \ x \ --> infinity such that X does not become extinct if beta is above this threshold, whereas it does below the threshold (where for this case beta 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 beta is no longer bounded from above and, in fact, degenerates to ka single point source delta(0). In this case, its expectation grows exponentially as t --> infinity. (C) 2000 Published by Elsevier Science B.V. All rights reserved. MSC: Primary 60J80, secondary 60J60; 60G57.