Iterating Brownian Motions, Ad Libitum

Nicolas Curien () and Takis Konstantopoulos ()
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Nicolas Curien: École Normale Supérieure
Takis Konstantopoulos: Uppsala University

Journal of Theoretical Probability, 2014, vol. 27, issue 2, 433-448

Abstract: Abstract Let B 1,B 2,… be independent one-dimensional Brownian motions parameterized by the whole real line such that B i (0)=0 for every i≥1. We consider the nth iterated Brownian motion W n (t)=B n (B n−1(⋯(B 2(B 1(t)))⋯)). Although the sequence of processes (W n ) 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 W n 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 ∞.

Keywords: Brownian motion; Iterated Brownian motion; Harris chain; Random measure; Exchangeability; Weak convergence; Local time; de Finetti–Hewitt–Savage theorem; 60J65; 60J05; 60G57; 60E99 (search for similar items in EconPapers)
Date: 2014
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DOI: 10.1007/s10959-012-0434-3

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