Measure-Valued Branching Diffusions with Singular Interactions

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Abstract
The usual super-Brownian motion is a measure-valued process that arises as a high density limit of a system of branching Brownian particles in which the branching mechanism is critical. In this work we consider analogous processes that model the evolution of a system of two such populations in which there is inter-species competition or predation.We first consider a competition model in which inter-species collisions may result in casualties on both sides. Using a Girsanov approach, we obtain existence and uniqueness of the appropriate martingale problem in one dimension. In two and three dimensions we establish existence only. However, we do show that, in three dimensions, any solution will not be absolutely continuous with respect to the law of two independent super-Brownian motions. Although the supports of two independent super-Brownian motions collide in dimensions four and five, we show that there is no solution to the martingale problem in these cases.We next study a prédation model in which collisions only affect the "prey" species. Here we can show both existence and uniqueness in one, two and three dimensions. Again, there is no solution in four and five dimensions. As a tool for proving uniqueness, we obtain a representation of martingales for a super-process as stochastic integrals with respect to the related orthogonal martingale measure.We also obtain existence and uniqueness for a related single population model in one dimension in which particles are killed at a rate proportional to the local density. This model appears as a limit of a rescaled contact process as the range of interaction goes to infinity.

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