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The Brownian web (BW), which developed from the work of Arratia and thenT\'{o}th and Werner, is a random collection of paths (with specified startingpoints) in one plus one dimensional space-time that arises as the scaling limitof the discrete web (DW) of coalescing simple random walks. Two recentlyintroduced extensions of the BW, the Brownian net (BN) constructed by Sun andSwart, 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 discreteextensions have a natural geometric structure in which the underlying Bernoullileft 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 meansof 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 wherearrow direction is replaced by the left or right parity of the (1,2) space-timepoints of the BW (points with one incoming path from the past and two outgoingpaths 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 alternateconstruction of the BN to that of Sun and Swart by proving that the switchingor branching can be implemented by a Poissonian marking of the (1,2) points.

Marking (1,2) Points of the Brownian Web and Applications