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Iterating Brownian motions, ad libitum. (English) Zbl 1331.60159
The \(n\)-fold iterated Brownian motion \(W_n\) is defined by \[ W_n(t) = B_n(B_{n-1}(\dots (B_2(B_1(t))) \dots)), \] where \(B_1,B_2,\dots\) are independent two-sided standard Brownian motions, such that \(B_n(0)=0\) for all \(n \geq 1\). K. Burdzy [Prog. Probab. 33, 67–87 (1992; Zbl 0789.60060)] has previously studied the sample-path properties of this process in the case \(n=2\). The present paper explores the asymptotics of the process \(W_n\) as \(n \rightarrow \infty\).
As the authors explain, the process \(W_n\) is not a semimartingale unless \(n=1\) and increasing \(n\) roughens the sample paths of the process (\(W_n\) can be shown to have finite \(2^n\)-variation). Thus, it is not obvious that \(W_n\) has a limit as \(n \rightarrow \infty\) and, indeed, the question turns out to be delicate. The first main result of the paper shows that any finite-dimensional distribution of \(W_n\) converges weakly to an exchangeable distribution. However, weak convergence of the process \(W_n\) in any reasonable function space seems impossible and, in fact, it is shown in the paper that the laws of \(W_n\), \(n \geq 1\), are not tight in the space of continuous paths, equipped with the locally uniform topology. The proof of the convergence of the finite-dimensional distributions of \(W_n\) is based on a nice argument that treats \((W_n(x_1),\dots,W_n(x_p))\), for distinct nonzero real numbers \(x_1,\dots,x_p\), as a Markov chain parameterized by \(n\), which is shown to be positive Harris recurrent.
The second main result of the paper concerns the occupation measure \(\mu_n\) of \(W_n\), defined via \[ \int_{\mathbb R} f\, d \mu_n = \int_0^1 f(W_n(t))\, dt \] for any Borel function \(f: {\mathbb R} \rightarrow [0,\infty)\). The authors show that the random measure \(\mu_n\) converges, as \(n \rightarrow \infty\), to a random measure \(\mu_\infty\), which is absolutely continuous with respect to the Lebesgue measure almost surely. This result parallels the principle, suggested by Simeon Berman, that very rough functions should have smooth local times.

60J65 Brownian motion
60J05 Discrete-time Markov processes on general state spaces
60F05 Central limit and other weak theorems
60G57 Random measures
60G09 Exchangeability for stochastic processes
60J55 Local time and additive functionals
60E99 Distribution theory
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