Iterating Brownian motions, ad libitum

Let B1, B2, ... be independent one-dimensional Brownian motions defined over the whole real line such that Bi(0) = 0 for every i ≥ 1. We consider the nth iterated Brownian motion Wn(t) = Bn(Bn−1(...(B2(B1(t)))...)). Although the sequences of processes (Wn)n≥1 do 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 Wn converge towards a random probability measure μ∞. We then prove that μ∞ almost surely has a continuous density which must be thought of as the local time process of the infinite iteration of independent Brownian motions.