Iterating Brownian Motions, Ad Libitum

作者：Curien Nicolas^*; Konstantopoulos Takis

来源：Journal of Theoretical Probability, 2014, 27(2): 433-448.

DOI：10.1007/s10959-012-0434-3

• Brownian motion
• Iterated Brownian motion
• Harris chain
• Random measure
• Exchangeability
• Weak convergence
• Local time
• de Finetti-Hewitt-Savage theorem

Let B-1, B-2, aEuro broken vertical bar be independent one-dimensional Brownian motions parameterized by the whole real line such that B (i) (0)=0 for every ia parts per thousand yen1. We consider the nth iterated Brownian motion W (n) (t)=B (n) (B (n-1)(a <-(B (2)(B (1)(t)))a <-)). Although the sequence of processes (W (n) ) (na parts per thousand yen1) 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 mu (a). We then prove that mu (a) almost surely has a continuous density which should be thought of as the local time process of the infinite iteration W (a) of independent Brownian motions. We also prove that the collection of random variables (W (a)(t),taa"ea-{0}) is exchangeable with directing measure mu(infinity).

• 出版日期2014-6