Weak convergence of the localized disturbance flow to the coalescing Brownian flow
Weak convergence of the localized disturbance flow to the coalescing Brownian flow
复制标题
局部扰动流向合并布朗流的弱收敛
DOI：
10.1214/13-aop845
复制
发表时间：
2011
影响因子：
2.3
通讯作者：
Amanda G. Turner
中科院分区：
文献类型：
--
作者：
J. Norris;Amanda G. Turner
我们在圆上为合并布朗流（也称为布朗网）定义了一个新的状态空间。该空间的元素是圆的保序映射族，连续依赖于两个时间参数，并具有一定的弱流动性质。该空间具有完全可分离度量。还引入了一个更大的状态空间，允许时间跳跃，并配备了一个skorokhod型度量，也是完全和可分离的。我们证明了合并布朗流是一类由跳跃演化而来的流族在这个较大空间中的弱极限，每个跳跃都是由圆的局部扰动引起的。本文还得到了这一结果的一个局部版本，其中弱极限律是线上合并布朗流的极限律。我们的建立很好地适应了时间反转，我们的弱极限结果为合并布朗流的时间可逆性提供了一个新的证明。我们还确定了一个与圆上合并布朗流相关的鞅，并利用它直接计算完成合并时间的拉普拉斯变换。
We define a new state-space for the coalescing Brownian flow, also known as the Brownian web, on the circle. The elements of this space are families of order-preserving maps of the circle, depending continuously on two time parameters and having a certain weak flow property. The space is equipped with a complete separable metric. A larger state-space, allowing jumps in time, is also introduced, and equipped with a Skorokhod-type metric, also complete and separable. We prove that the coalescing Brownian flow is the weak limit in this larger space of a family of flows which evolve by jumps, each jump arising from a small localized disturbance of the circle. A local version of this result is also obtained, in which the weak limit law is that of the coalescing Brownian flow on the line. Our set-up is well adapted to time-reversal and our weak limit result provides a new proof of time-reversibility of the coalescing Brownian flow. We also identify a martingale associated with the coalescing Brownian flow on the circle and use this to make a direct calculation of the Laplace transform of the time to complete coalescence.