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Measure-valued branching diffusions with singular interactions


© Canadian Mathematical Society 1994. The usual super-Brownian motion is a measure-valued process that arises as a high density limit of a system of branching Brownian partitles in which the branching methanism 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 prédation. We first considera competition model in which inter-species collisions may result in casualties on both sides, Using a Girsanov approach, we oblain 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 he 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 lour and live, we show that there is no solution lo 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 live dimensions. As a tool for proving uniqueness, we obtain a representation of martingales for a super-proeessas stochastic integrals with respect to the related orthogonal martingale measure. W also obtain existence and uniqueness for a related single population model in one dimension in which panicles are killed at a rate proportional to Ihe local densily. This model appears as a limit oía rescaled contati process as the range (if interaction goes to infinity.

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