Probability Theory

  • Horst Osswald
Part of the Mathematics and Its Applications book series (MAIA, volume 510)


In this chapter we apply the Saturation Principle to the profound mathematical theory of Brownian motion and, ultimately, analysis on Wiener spaces. The latter subject is often called the Malliavin-calculus or the stochastic calculus of variation. Using the *-extension of the positive integers, the notion of finiteness can be extended so that processes with discrete time and those with continuous time are almost indistinguishable. Moreover, the L p -space over Wiener measure on the set C [0,1] of continuous functions on [0,1] is equivalent to a subspace of L p (*ℝ H L ). Here H is an unlimited positive integer and Γ L is the Loeb measure — over the H-fold centered Gaussian distribution with variance 1/H — restricted to the σ-algebra generated by a Brownian motion defined on the hyperfinite dimensional Euclidean space * H Equivalence of these Banach spaces means that analysis on the classical Wiener space is closely related to the finite dimensional calculus. The idea of taking *ℝ H instead of C[0,1] is due to Nigel Cutland and Siu A-Ng [7].


Brownian Motion Standard Part Wiener Space Admissible Sequence Internal Function 
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Copyright information

© Springer Science+Business Media Dordrecht 2000

Authors and Affiliations

  • Horst Osswald
    • 1
  1. 1.Mathematisches Institut der Universität MünchenMünchenGermany

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