Change of variable formulas for infinite-dimensional distributions. (English. Russian original) Zbl 0898.46043

Math. Notes 60, No. 2, 212-215 (1996); translation from Mat. Zametki 60, No. 2, 288-292 (1996).
The main examples of objects to which the results presented below can be applied are countably additive smooth measures on (infinite-dimensional) locally convex spaces, measures known as the Feynman measures and some other functional integrals, as well as uncountably additive measures, including those generated by Lévy’s Laplacians. We shall consider a general method of deriving formulas that describe the transformations of distributions (generalized measures) on locally convex spaces under (nonlinear) smooth transformations of these spaces. In particular, this method yields analogs (in the case of distributions) of the familiar formulas of Cameron-Martin, Girsanov-Maruyama, and Reimer. The method was used to investigate transformations of smooth measures and to study transformations of the Feynman measure.
In addition to the description of the general method that covers all the cases listed above, the note also contains a number of new specific formulas. It should be emphasized that, as distinct from all the works mentioned above, we do not restrict ourselves to the mapping of locally convex spaces that can be connected to the identity mapping by a continuous curve (in the space of mappings). To achieve this extension of the applicability area of the method, we pass from real locally convex spaces to their complexifications.


46F25 Distributions on infinite-dimensional spaces
46G12 Measures and integration on abstract linear spaces
46F10 Operations with distributions and generalized functions
28C20 Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.)
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