Weak convergence of the localized disturbance flow to the coalescing Brownian flow

Details Weak convergence of the localized disturbance flow to the coalescing Brownian flow Weak convergence of the localized disturbance flow to the coalescing Brownian flow

2011-06-16

arxiv.org/abs/1106.3252v3

Mathematics

Probability

37 pages. arXiv admin note: substantial text overlap with arXiv:0810.0211

Scientific paper

We define a new state-space for the coalescing Brownian flow on the circle. This space is a complete separable metric space of maps on the circle with a certain weak flow property and having continuous time-dependence. A larger state-space, allowing jumps in time, is also introduced, and equipped with a Skorokhod-type metric. We prove that the coalescing Brownian flow is the weak limit in this larger space of a family of discrete-time flows generated by small localized disturbances 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. We finally explore the relationship between our formulation of the coalescing Brownian flow and the Brownian web.

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