论文标题
与成对相互作用的分支过程
Branching processes with pairwise interactions
论文作者
论文摘要
在此手稿中，我们对成对相互作用（BPI-Processes）的分支过程的长期行为感兴趣。该类别的过程表现为一个纯粹的分支过程，其差异也允许成对之间的竞争和合作事件。在这里，我们提供了一系列积分测试，以解释竞争与合作如何调节BPI-Processes的长期行为。特别是，这种整体测试描述了爆炸和灭绝的事件，并提供了该过程降低无穷大的条件。此外，我们还确定该过程是否承认固定分布。我们的论点根据修改后的分支过程，使用随机的时间变化表示，并具有移民和二元性。 BPI-Processes的双重偶数是一个散布的家族，以$ [0,1] $的价值，这本身就是有趣的，我们将其作为广义的Wright-Fisher扩散介绍。
In this manuscript, we are interested in the long-term behaviour of branching processes with pairwise interactions (BPI-processes). A process in this class behaves as a pure branching process with the difference that competition and cooperation events between pairs of individuals are also allowed. Here, we provide a series of integral tests that explain how competition and cooperation regulate the long-term behaviour of BPI-processes. In particular, such integral tests describe the events of explosion and extinction and provide conditions under which the process comes down from infinity. Moreover, we also determine whether the process admits, or not, a stationary distribution. Our arguments use a random time change representation in terms of a modified branching process with immigration and moment duality. The moment dual of BPI-processes turns out to be a family of diffusions taking values on $[0,1]$ which are interesting in their own right and that we introduce as generalised Wright-Fisher diffusions.
