Markov branching diffusions: martingales, Girsanov type theorems and applications to the long term behaviour

Consider a spatial branching particle process where the underlying motion is a conservative diffusion on D C Rd corresponding to the elliptic op- erator L on D, and the branching is strictly binary (dyadic), with spatially varying rate ß(x) => 0 (and ß 0) which is assumed to be bounded from above. We prove that, under extremely mild circumstances the process exhibits local extinction if and only if ac <= 0, where ac denotes the gen- eralized principal eigenvalue for the operator L + ß on D. (This criterion is analogous to the one obtained by Pinsky (1996) for the local extinction of superdiffusions). Furthermore we show that when the process does not exhibit local extinction, every nonempty open subset is occupied infinitely often with positive probability which can be characterized by a solution bounded in (0,1] to the semilinear elliptic equation Lu + ß(u² - u) = 0 on D. Moreover, in this case, there is an exponential rate of growth on suficiently large compact domains, and this rate can be arbitrarily close to ac. In order to reach these conclusions we first develop some results concerning innerproduct and multiplicative martingales and their relation to the operators L + ß and L + ßv respectively, where v (x) = x² - x. In the case of the innerproduct martingales we show that for some cir- cumstances they can be used as changes of measure for the law of the branching process in a similar way that Girsanov densities act as changes of measure in the context of diffusions. More specifically, the change of measure induces a drift consistent with a certain Doob's h-transform on the path of a randomized ancestral line of descent. These concepts are essentially spatial versions of spine decompositions for Galton-Watson processes given in Lyons et al. (1995)