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Transformation of Wiener measure under anticipative flows. (English) Zbl 0767.60046
Summary: Let \(T(\omega)=\omega+F(\omega)\) be a transformation from the Wiener space to itself with the range of \(F(\omega)\) assumed to be in the Cameron-Martin space. The absolute continuity and the density function associated with \(T\) is considered; \(T\) is assumed to be embedded in or defined through a parameterization \(T_ t\omega=\omega+F_ t(\omega)\) and \(F_ t\) is assumed to be differentiable in \(t\). The paper deals first with the case where the range of the \(t\)-derivative of \(F_ t(\omega)\) is also in the Cameron-Martin space and new representations for the Radon-Nikodym derivative and the Carleman-Fredholm determinant are derived. The case where the \(t\)-derivative of \(F_ t\) is not in the Cameron-Martin space is considered next and results on the absolute continuity and the density function, under conditions which are considerably weaker than previously known conditions, are presented.

60H05 Stochastic integrals
60G20 Generalized stochastic processes
60G17 Sample path properties
Full Text: DOI
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