Some Path Properties of Iterated Brownian Motion

Abstract

Suppose that X¹, X² and Y are independent standard Brownian motions starting from 0 and let

$$ X\left( t \right) = \left\{ {\begin{array}{*{20}{c}} {{X^1}\left( t \right) if t \geqslant 0,} \\ {{X^2}\left( { - t} \right) if t < 0.} \end{array}} \right.$$

We will consider the process $\left\{ {Z\left( t \right)\underline{\underline {df}} X\left( {Y\left( t \right)} \right),t \geqslant 0} \right\}$ which we will call “iterated Brownian motion” or simply IBM. Funaki (1979) proved that a similar process is related to “squared Laplacian.” Krylov (1960) and Hochberg (1978) considered finitely additive signed measures on the path space corresponding to squared Laplacian (there exists a genuine probabilistic approach, see, e.g., Mądrecki and Rybaczuk (1992). A paper of Vervaat (1985) contains a section on the composition of self-similar processes.