Iterating Brownian Motions,Ad Libitum

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∈??{0}) is exchangeable with directing measure μ _∞.