Iterating Brownian motions, ad libitum

TL;DR

This paper studies the behavior of iterated Brownian motions, showing that while the processes do not converge functionally, their finite-dimensional distributions and occupation measures do, revealing a continuous density linked to local time.

Contribution

It establishes convergence of finite-dimensional marginals and occupation measures for iterated Brownian motions, introducing a new perspective on their limiting behavior.

Findings

Finite-dimensional marginals of iterated Brownian motions converge.

Occupation measures of iterated Brownian motions converge to a random measure.

The limiting measure has a continuous density related to local time.

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 \mu_\infty. We then prove that \mu_\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.