Iterating Brownian Motions, Ad Libitum

Abstract

Let B 1,B 2,… be independent one-dimensional Brownian motions parameterized by the whole real line such that B i (0)=0 for every i≥1. We consider the nth iterated Brownian motion W n (t)=B n (B n−1(⋯(B 2(B 1(t)))⋯)). Although the sequence of processes (W n ) n≥1 does not converge in a functional sense, we prove that the finite-dimensional marginals converge. As a consequence, we deduce that the random occupation measures of W n converge to a random probability measure μ ∞. We then prove that μ ∞ almost surely has a continuous density which should be thought of as the local time process of the infinite iteration W ∞ of independent Brownian motions. We also prove that the collection of random variables (W ∞(t),t∈ℝ∖{0}) is exchangeable with directing measure μ ∞.