# Some Path Properties of Iterated Brownian Motion

• Krzysztof Burdzy
Chapter
Part of the Progress in Probability book series (PRPR, volume 33)

## 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.

## Keywords

Brownian Motion Local Maximum Quadratic Variation Path Space Path Property
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## References

1. 1.
T. Funaki, Probabilistic construction of the solution of some higher order parabolic differential equations, Proc. Japan Acad. 55 (1979), 176–179.
2. 2.
K.J. Hochberg, A signed measure on path space related to Wiener measure, Ann. Probab. 6 (1978), 433–458.
3. 3.
I.Karatzas and S. Shreve, Brownian Motion and Stochastic Calculus, Springer, New York, 1988.
4. 4.
V. Yu. Krylov, Some properties of the distribution corresponding to the equationu/∂t=(−1)q+12q u/∂x 2q, Soviet Math. Dokl. 1 (1960), 760–763.
5. 5.
A. Mądrecki and M. Rybaczuk, New Feynman-Kac type formula, (preprint) (1992).Google Scholar
6. 6.
W. VervaatSample path properties of self-similar processes with stationary increments, Ann. Probab. 13 (1985), 1–27.