Iterating Brownian Motions, Ad Libitum

Research paper by Nicolas Curien, Takis Konstantopoulos

Indexed on: 19 Jul '12Published on: 19 Jul '12Published in: Journal of Theoretical Probability


Let B1,B2,… be independent one-dimensional Brownian motions parameterized by 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 sequence of processes (Wn)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 Wn 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∞.