2011/12/16 by Nicolas Curien, Curien, Nicolas, Takis Konstantopoulos +1
Economics, Econometrics and Finance · Mathematics · #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Stochastic processes and financial applications #Stochastic processes and statistical mechanics #math.PR
paper · pdf · doi:10.48550/arxiv.1112.3776
arxiv created 2011/12/16 · openalex publication_date 2011/12/16 · arxiv updated 2011/12/19 · openalex created_date 2022/10/04 · openalex updated_date 2026/07/28
Let B1,B2, ... be independent one-dimensional Brownian motions defined over the whole real line such that Bi(0)=0. We consider the nth iterated Brownian motion Wn(t)= Bn(Bn-1(...(B2(B1(t)))...)). Although the sequences of processes (Wn) 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 Wn converge towards a random probability measure μ_∞. We then prove that μ_∞ almost surely has a continuous density which must be thought of as the local time process of the infinite iteration of independent Brownian motions.