Radon-Nikodym Derivatives of Finitely Additive Measures Induced by Nonlinear Transformations on Hilbert Space

摘要

In the paper the authors study the issue of evaluating the Radon-Nikodym derivatives of finitely additive measures induced by nonlinear transformations of the type (I - M)(sup -1) on a Hilbert space H with standard Gauss measure thereon. Under suitable conditions on the nonlinear map M, the authors show that the R-N derivative is given by a Jacobi type transformation. The authors then evaluate the R-N derivative for the process which satisfies a nonlinear differential equation driven by additive white noise (which is the identity map on H under Gauss measure). In particular it is shown that this is the white noise analog of the Cameron-Martin formula with an additional term which when interpreted for paths in C(0,T) corresponds to the Wong-Zakai correction term.