The Noise Made by a Poisson Snake

Abstract

The purpose of this article is to study a coalescing flow of sticky Brownian motions. Sticky Brownian motion arises as a weak solution of a stochastic differential equation, and the study of the flow reveals the nature of the extra randomness that must be added to the driving Brownian motion. This can be represented in terms of Poissonian marking of the trees associated with the excursions of Brownian motion. We also study the noise, in the sense of Tsirelson, generated by the flow. It is shown that this noise is not generated by any Brownian motion, even though it is predictable.