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    Rights statement: © Institute of Mathematical Statistics, 2015

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

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Weak convergence of the localized disturbance flow to the coalescing Brownian flow. / Norris, James; Turner, Amanda.
In: Annals of Probability, Vol. 43, No. 3, 05.05.2015, p. 935-970.

Research output: Contribution to Journal/Magazine › Journal article › peer-review

Harvard

Norris, J & Turner, A 2015, 'Weak convergence of the localized disturbance flow to the coalescing Brownian flow', Annals of Probability, vol. 43, no. 3, pp. 935-970. https://doi.org/10.1214/13-AOP845

APA

Norris, J., & Turner, A. (2015). Weak convergence of the localized disturbance flow to the coalescing Brownian flow. Annals of Probability, 43(3), 935-970. https://doi.org/10.1214/13-AOP845

Vancouver

Norris J, Turner A. Weak convergence of the localized disturbance flow to the coalescing Brownian flow. Annals of Probability. 2015 May 5;43(3):935-970. doi: 10.1214/13-AOP845

Author

Norris, James ; Turner, Amanda. / Weak convergence of the localized disturbance flow to the coalescing Brownian flow. In: Annals of Probability. 2015 ; Vol. 43, No. 3. pp. 935-970.

Bibtex

@article{8ee3d7b757c244adb13965a8720fc2db,
title = "Weak convergence of the localized disturbance flow to the coalescing Brownian flow",
abstract = "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.",
keywords = "stochastic flow, coalescing Brownian motions, Brownian web, Arratia flow",
author = "James Norris and Amanda Turner",
note = " {\textcopyright} Institute of Mathematical Statistics, 2015",
year = "2015",
month = may,
day = "5",
doi = "10.1214/13-AOP845",
language = "English",
volume = "43",
pages = "935--970",
journal = "Annals of Probability",
publisher = "Institute of Mathematical Statistics",
number = "3",

}

RIS

TY - JOUR

T1 - Weak convergence of the localized disturbance flow to the coalescing Brownian flow

AU - Norris, James

AU - Turner, Amanda

N1 - © Institute of Mathematical Statistics, 2015

PY - 2015/5/5

Y1 - 2015/5/5

N2 - 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.

AB - 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.

KW - stochastic flow

KW - coalescing Brownian motions

KW - Brownian web

KW - Arratia flow

U2 - 10.1214/13-AOP845

DO - 10.1214/13-AOP845

M3 - Journal article

VL - 43

SP - 935

EP - 970

JO - Annals of Probability

JF - Annals of Probability

IS - 3

ER -