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 n th 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 ∞ .