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.
Keywords:
Cameron-Martin space; new representations for the Radon-Nikodym derivative; absolute continuityReferences:
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