Nonlinear transformations of the canonical gauss measure on hilbert space and absolute continuity

Abstract

The papers of R. Ramer and S. Kusuoka investigate conditions under which the probability measure induced by a nonlinear transformation on abstract Wiener space(γ ,H,B) is absolutely continuous with respect to the abstract Wiener measure μ. These conditions reveal the importance of the underlying Hilbert space H but involve the space B in an essential way. The present paper gives conditions solely based on H and takes as its starting point, a nonlinear transformation T=I+F on H. New sufficient conditions for absolute continuity are given which do not seem easily comparable with those of Kusuoka or Ramer but are more general than those of Buckdahn and Enchev. The Ramer-Ito integral occurring in the expression for the Radon-Nikodym derivative is studied in some detail and, in the general context of white noise theory it is shown to be an anticipative stochastic integral which, under a stronger condition on the weak Gateaux derivative of F is directly related to the Ogawa integral.