Transformations of order one and quadratic forms on Wiener spaces
Setsuo Taniguchi
TL;DR
The paper establishes a bidirectional correspondence between transformations of order one on Wiener spaces and quadratic forms. It generalizes the finite-dimensional identity between a linear transform I+A and a quadratic exponent to the infinite-dimensional Wiener space via iota+F_kappa and the quadratic form q_eta(kappa), proving a change-of-variables formula under the condition Lambda(B_eta)<1. It also proves the existence of inverse order-one transformations and analyzes surjectivity/injectivity properties, including a bijection on a restricted class and Laplace-transform evaluations. Finally, it connects these results to Cameron–Martin linear transformations, providing explicit determinant expressions and demonstrating consistency with Malliavin calculus-enabled change-of-variables on Wiener space.
Abstract
It will be shown that transformations of order one on the Wiener space give rise to quadratic forms as exponents of change of variables formulas, and conversely every exponentially integrable quadratic form has a transformation of order one realizing the form in such a manner. Several expressions of corresponding change of variables formulas are also discussed.