Iterating Brownian motions, ad libitum

Abstract

Let B_1,B_2, ... be independent one-dimensional Brownian motions defined over the whole real line such that B_i(0)=0. We consider the nth iterated Brownian motion W_n(t)= B_n(B_{n-1}(...(B_2(B_1(t)))...)). Although the sequences of processes (W_n) 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 W_n converge towards a random probability measure μ_\infty. We then prove that μ_\infty almost surely has a continuous density which must be thought of as the local time process of the infinite iteration of independent Brownian motions.