Iterating Brownian motions, ad libitum
Probability
2011-12-19 v1
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.
Cite
@article{arxiv.1112.3776,
title = {Iterating Brownian motions, ad libitum},
author = {Nicolas Curien and Takis Konstantopoulos},
journal= {arXiv preprint arXiv:1112.3776},
year = {2011}
}