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Swart: The Brownian net
 GENERALIZED DRAINAGE NETWORK 21
, 2008
"... Abstract The (standard) Brownian web is a collection of coalescing onedimensional Brownian motions, starting from each point in space and time. It arises as the diffusive scaling limit of a collection of coalescing random walks. We show that it is possible to obtain a nontrivial limiting object if ..."
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Abstract The (standard) Brownian web is a collection of coalescing onedimensional Brownian motions, starting from each point in space and time. It arises as the diffusive scaling limit of a collection of coalescing random walks. We show that it is possible to obtain a nontrivial limiting object if the random walks in addition branch with a small probability. We call the limiting object the Brownian net, and study some of its elementary properties. MSC 2000. Primary: 82C21 ; Secondary: 60K35, 60F17, 60D05. Keywords. Brownian net, Brownian web, branchingcoalescing random walks, branchingcoalescing point set. Acknowledgements. Work sponsered by the DFG and by GAČR grant 201/06/1323. R. Sun thanksÚTIA and J.M. Swart thanks Eurandom for their hospitality during short visits, which were supported by RDSES short visit grants from the European Science Foundation. 1
Marking (1,2) Points of the Brownian Web and Applications
, 2009
"... The Brownian web (BW), which developed from the work of Arratia and then Tóth and Werner, is a random collection of paths (with specified starting points) in one plus one dimensional spacetime that arises as the scaling limit of the discrete web (DW) of coalescing simple random walks. Two recently ..."
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Cited by 6 (3 self)
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The Brownian web (BW), which developed from the work of Arratia and then Tóth and Werner, is a random collection of paths (with specified starting points) in one plus one dimensional spacetime that arises as the scaling limit of the discrete web (DW) of coalescing simple random walks. Two recently introduced extensions of the BW, the Brownian net (BN) constructed by Sun and Swart, and the dynamical Brownian web (DyBW) proposed by Howitt and Warren, are (or should be) scaling limits of corresponding discrete extensions of the DW — the discrete net (DN) and the dynamical discrete web (DyDW). These discrete extensions have a natural geometric structure in which the underlying Bernoulli left or right “arrow” structure of the DW is extended by means of branching (i.e., allowing left and right simultaneously) to construct the DN or by means of switching (i.e., from left to right and viceversa) to construct the DyDW. In this paper we show that there is a similar structure in the continuum where arrow direction is replaced by the left or right parity of the (1,2) spacetime points of the BW (points with one incoming path from the past and two outgoing paths to the future, only one of which is a continuation of the incoming path). We then provide a complete construction of
Marking (1,2) points of the Brownian web and applications1
, 2009
"... All intext references underlined in blue are linked to publications on ResearchGate, letting you access and read them immediately. ..."
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All intext references underlined in blue are linked to publications on ResearchGate, letting you access and read them immediately.
Brownian Net with Killing
, 2014
"... Abstract. Motivated by its relevance for the study of perturbations of onedimensional voter models, including stochastic Potts models at low temperature, we consider diffusively rescaled coalescing random walks with branching and killing. Our main result is convergence to a new continuum process, ..."
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Abstract. Motivated by its relevance for the study of perturbations of onedimensional voter models, including stochastic Potts models at low temperature, we consider diffusively rescaled coalescing random walks with branching and killing. Our main result is convergence to a new continuum process, in which the random spacetime paths of the SunSwart Brownian net are terminated at a Poisson cloud of killing points. We also prove existence of a percolation transition as the killing rate varies. Key issues for convergence are the relations of the discrete model killing points and their intensity measure to the continuum counterparts. 1 ar X iv
© Institute of Mathematical Statistics, 2008 THE BROWNIAN NET
"... The (standard) Brownian web is a collection of coalescing onedimensional Brownian motions, starting from each point in space and time. It arises as the diffusive scaling limit of a collection of coalescing random walks. We show that it is possible to obtain a nontrivial limiting object if the rando ..."
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The (standard) Brownian web is a collection of coalescing onedimensional Brownian motions, starting from each point in space and time. It arises as the diffusive scaling limit of a collection of coalescing random walks. We show that it is possible to obtain a nontrivial limiting object if the random walks in addition branch with a small probability. We call the limiting object the Brownian net, and study some of its elementary properties. Contents
Marking (1,2) Points of the Brownian Web and Applications
, 2009
"... The Brownian web (BW), which developed from the work of Arratia and then Tóth and Werner, is a random collection of paths (with specified starting points) in one plus one dimensional spacetime that arises as the scaling limit of the discrete web (DW) of coalescing simple random walks. Two recently ..."
Abstract
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The Brownian web (BW), which developed from the work of Arratia and then Tóth and Werner, is a random collection of paths (with specified starting points) in one plus one dimensional spacetime that arises as the scaling limit of the discrete web (DW) of coalescing simple random walks. Two recently introduced extensions of the BW, the Brownian net (BN) constructed by Sun and Swart, and the dynamical Brownian web (DyBW) proposed by Howitt and Warren, are (or should be) scaling limits of corresponding discrete extensions of the DW — the discrete net (DN) and the dynamical discrete web (DyDW). These discrete extensions have a natural geometric structure in which the underlying Bernoulli left or right “arrow ” structure of the DW is extended by means of branching (i.e., allowing left and right simultaneously) to construct the DN or by means of switching (i.e., from left to right and viceversa) to construct the DyDW. In this paper we show that there is a similar structure in the continuum where arrow direction is replaced by the left or right parity of the (1,2) spacetime points of the BW (points with one incoming path from the past and two outgoing paths to the future, only one of which is a continuation of the incoming path). We then provide a complete construction of the DyBW and an alternate construction of the BN to that of Sun and Swart by proving that the switching or branching