Measure-Valued Branching Diffusions with Singular Interactions
Measure-Valued Branching Diffusions with Singular Interactions
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DOI：
10.4153/cjm-1994-004-6
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发表时间：
1994-02
期刊：
影响因子：
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通讯作者：
S. Evans;E. Perkins
中科院分区：
文献类型：
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作者：
S. Evans;E. Perkins
摘要通常的超布朗运动是一个测值过程，它是作为一个分支布朗粒子系统的高密度极限而产生的，其中分支机制是关键的。在这项工作中，我们考虑了模拟两个这样的种群系统的演化的相似过程，在这个系统中，存在物种间的竞争或捕食。我们首先考虑了一个竞争模型，其中物种间的碰撞可能会导致双方的伤亡。利用Girsanov方法，我们得到了一维适当的鞅问题的存在唯一性。在二维和三维空间中，我们只确立了存在。然而，我们确实证明了，在三维空间中，关于两个独立的超布朗运动定律，任何解都不是绝对连续的。虽然两个独立的超布朗运动的支撑点在四维和五维上相撞，但我们证明了在这些情况下，鞅问题是没有解的。接下来，我们研究一个碰撞只影响“猎物”物种的捕食模型。在这里，我们可以在一维、二维和三维中显示存在和唯一性。同样，在四个和五个维度上都没有解决方案。作为一种证明唯一性的工具，我们得到了超过程的随机积分的一个表示形式，即关于相关的正交鞅测度的随机积分。我们还得到了一维相关的单种群模型的存在性和唯一性，其中粒子以与局部密度成正比的速度被杀死。随着相互作用的范围变得无限大，这个模型看起来就像是重新调整的接触过程的极限。
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.