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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 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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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.MathSciNetMATHCrossRefGoogle Scholar
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    K.J. Hochberg, A signed measure on path space related to Wiener measure, Ann. Probab. 6 (1978), 433–458.MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    I.Karatzas and S. Shreve, Brownian Motion and Stochastic Calculus, Springer, New York, 1988.MATHCrossRefGoogle Scholar
  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.MathSciNetMATHGoogle Scholar
  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.MathSciNetMATHCrossRefGoogle Scholar

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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