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May 2015 Weak convergence of the localized disturbance flow to the coalescing Brownian flow
James Norris, Amanda Turner
Ann. Probab. 43(3): 935-970 (May 2015). DOI: 10.1214/13-AOP845


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.


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James Norris. Amanda Turner. "Weak convergence of the localized disturbance flow to the coalescing Brownian flow." Ann. Probab. 43 (3) 935 - 970, May 2015.


Published: May 2015
First available in Project Euclid: 5 May 2015

zbMATH: 1327.60086
MathSciNet: MR3342655
Digital Object Identifier: 10.1214/13-AOP845

Primary: 60F17

Keywords: Arratia flow , Brownian web , Coalescing Brownian motions , stochastic flow

Rights: Copyright © 2015 Institute of Mathematical Statistics


Vol.43 • No. 3 • May 2015
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