Levy Downcrossing Theorem for the Arratia Flow

Abstract

We study the total number of downcrossings of a fixed strip by the trajectories of a continuum system of particles from the Arratia flow. We prove the convergence of the product of the strip width by the total number of downcrossings of the strip to the total local time for the Arratia flow. This statement is an analog of the well-known Levy downcrossing theorem for a Wiener process.