Some Path Properties of Iterated Brownian Motion

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


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.


Brownian Motion Local Maximum Quadratic Variation Path Space Path Property 
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Copyright information

© Springer Science+Business Media New York 1993

Authors and Affiliations

  • Krzysztof Burdzy
    • 1
  1. 1.Department of MathematicsUniversity of WashingtonSeattleUSA

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