SMILE

Stochastic Models for the Inference of Life Evolution

Marking (1, 2) points of the Brownian web and applications

Newman, C., Ravishankar,, K., Schertzer, E.

Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

2010

The Brownian web (BW), which developed from the work of Arratia and then T\'oth and Werner, is a random collection of paths (with specified starting points) in one plus one dimensional space-time 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 vice-versa) 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) space-time 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 can be implemented by a Poissonian marking of the (1, 2) points.

Bibtex

@article{newman_marking_2010,
Author = {Newman, CM and Ravishankar,, K and Schertzer, E},
Title = {Marking (1, 2) points of the {Brownian} web and
applications},
Journal = {Annales de l'Institut Henri Poincaré, Probabilités
et Statistiques},
Volume = {46},
Number = {2},
Pages = {537--574.},
abstract = {The Brownian web (BW), which developed from the work
of Arratia and then T\'oth and Werner, is a random
collection of paths (with specified starting points) in
one plus one dimensional space-time 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 vice-versa) 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) space-time 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 can be implemented by a Poissonian marking of
the (1, 2) points. },
doi = {10.1214/09-AIHP325},
shorttitle = {Ann. {Inst}. {H}. {Poincaré} {Probab}. {Statist}.},
year = 2010
}

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