ON THE BROWNIAN NET AND THE HOWITT-WARREN FLOWS

Abstract

The Brownian net is a collection of branching-coalescing Brownian motions starting from every point in the space-time plane R^2, which has been shown to be the diffusive scaling limit of branching-coalescing simple random walks, where the random walk paths do not cross. The first part of the thesis is devoted to showing that the Brownian net is contained in any subsequential weak limit of nonsimple random walks with crossing paths. In the second part, we study current fluctuations in a one-dimensional interacting particle system known as the dual smoothing process that is dual to random motions in a Howitt-Warren flow. The Howitt-Warren flow can be regarded as the family of transition kernels of a random motion in a continuous space-time random environment, which can be constructed from the Brownian web and net. It turns out that the current fluctuations fall in the Edwards-Wilkinson universality class, where the fluctuations occur on the scale t^{1/4} and the limit is a universal Gaussian process.