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Iterating Brownian Motions, Ad Libitum

Author

Listed:
  • Nicolas Curien

    (École Normale Supérieure)

  • Takis Konstantopoulos

    (Uppsala University)

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 ∞.

Suggested Citation

  • Nicolas Curien & Takis Konstantopoulos, 2014. "Iterating Brownian Motions, Ad Libitum," Journal of Theoretical Probability, Springer, vol. 27(2), pages 433-448, June.
    • Handle: RePEc:spr:jotpro:v:27:y:2014:i:2:d:10.1007_s10959-012-0434-3
    DOI: 10.1007/s10959-012-0434-3
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    References listed on IDEAS

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    1. Bertoin, Jean, 1996. "Iterated Brownian motion and stable() subordinator," Statistics & Probability Letters, Elsevier, vol. 27(2), pages 111-114, April.
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