论文标题
持续状态分支过程与碰撞：第一通道时间和二元性
Continuous-state branching processes with collisions: first passage times and duality
论文作者
论文摘要
我们通过考虑到随机碰撞的现象来介绍一类一维积极的马尔可夫过程（CBS）。除分支外，以一般机制$ψ$为特征的分支，以恒定的时间速率，两个粒子在人群中均匀地采样，碰撞并留下由（子）关键机制$σ$控制的大量颗粒。这种具有碰撞（CBC）的CB过程被证明是唯一没有负跳跃的候选者过程，使Laplace双重性关系与一维扩散在半线上。这概括了福卡（Foucart）观察到的逻辑CBS的二元性。通过时间变化，CBC也与Markov过程的辅助类别有关，该过程称为CB过程，该过程具有光谱正迁移（CBM），该过程最近由Vidmar引入。我们发现边界的必要条件$ 0 $或$ \ infty $要吸引和限制分配。提供了后者的拉普拉斯变换。在CBC过程没有爆炸的假设下，在二阶微分方程的解决方案的帮助下，第一个段落时间的拉普拉斯转换是任意水平以下的，其系数在lévy-khintchine函数$σ$和$ψ$方面表示。给出了足够的非探索条件。
We introduce a class of one-dimensional positive Markov processes generalizing continuous-state branching processes (CBs), by taking into account a phenomenon of random collisions. Besides branching, characterized by a general mechanism $Ψ$, at a constant rate in time two particles are sampled uniformly in the population, collide and leave a mass of particles governed by a (sub)critical mechanism $Σ$. Such CB processes with collisions (CBCs) are shown to be the only Feller processes without negative jumps satisfying a Laplace duality relationship with one-dimensional diffusions on the half-line. This generalizes the duality observed for logistic CBs by Foucart. Via time-change, CBCs are also related to an auxiliary class of Markov processes, called CB processes with spectrally positive migration (CBMs), recently introduced by Vidmar. We find necessary and sufficient conditions for the boundaries $0$ or $\infty$ to be attracting and for a limiting distribution to exist. The Laplace transform of the latter is provided. Under the assumption that the CBC process does not explode, the Laplace transforms of the first passage times below arbitrary levels are represented with the help of the solution of a second-order differential equation, whose coefficients express in terms of the Lévy-Khintchine functions $Σ$ and $Ψ$. Sufficient conditions for non-explosion are given.
