论文标题
$ n $ n $ urn分支过程的缩放限制和波动
Scaling limits and fluctuations of a family of $N$-urn branching processes
论文作者
论文摘要
在本文中，我们关注的是一个$ n $ urn的分支过程，其中一些粒子最初被放入$ n $ urns中，然后每个粒子在死亡时会在某些urn中生出几个新粒子。该型号包括$ n $ nur-urn ehrenfest模型和$ n $ urn-urn-urn分支随机步行作为特殊情况。我们表明，该过程的缩放限制是由$ c（\ mathbb {t}）$ - 有价值的线性的普通微分方程以及该过程的波动驱动的，由$ c^\ infty（\ nath ins $ c^\ ins $ c^\ ins $ c^wery is ins $ C^\ ins $ c = rnStectein-uhlenbeck ther一维的圆环。
In this paper we are concerned with a family of $N$-urn branching processes, where some particles are put into $N$ urns initially and then each particle gives birth to several new particles in some urn when dies. This model includes the $N$-urn Ehrenfest model and the $N$-urn branching random walk as special cases. We show that the scaling limit of the process is driven by a $C(\mathbb{T})$-valued linear ordinary differential equation and the fluctuation of the process is driven by a generalized Ornstein-Uhlenbeck process in the dual of $C^\infty(\mathbb{T})$, where $\mathbb{T}=(0, 1]$ is the one-dimensional torus. A crucial step for proofs of above main results is to show that numbers of particles in different urns are approximately independent. As applications of our main results, limit theorems of hitting times of the process are also discussed.
