Dynamics for the Brownian web and the erosion flow

Abstract

The Brownian web is a random object that occurs as the scaling limit of an infinite system of coalescing random walks. Perturbing this system of random walks by, independently at each point in space-time, resampling the random walk increments, leads to some natural dynamics. In this paper we consider the corresponding dynamics for the Brownian web. In particular, pairs of coupled Brownian webs are studied, where the second web is obtained from the first by perturbing according to these dynamics. A stochastic flow of kernels, which we call the erosion flow, is obtained via a filtering construction from such coupled Brownian webs, and the N-point motions of this flow of kernels are identified.