The Transformation of Wiener Integrals by Nonlinear Transformations
R. H. Cameron, W. T. Martin · Transactions of the American Mathematical Society · 1949
In this paper we study the behavior of Wiener integrals under transformations of the form (0.1) T: yit) = "(0 + A(_|i),where A(.v| t) is a functional depending on the function .%•and the number / and satisfying certain smoothness conditions.(The number / ranges over the interval 0 __ ¿ __ 1, and the function x ranges over a measurable subset V of the space C of continuous functions on 0 __ ¿ __ 1 which vanish when t = 0.) The smoothness conditions on A(#| /) require in particular that it have a Volterra derivative K~ix\t, s) such that d r1 (0.2) SA = -A(.r+ kSx \ t)]H=0 = I Kix \ t, j)5"(s)_s.dh JoThis Volterra derivative is somewhat analogous to the matrix of the partial derivatives of the n functions which define a transformation in n dimensions (or, more strictly, to this matrix minus the identity matrix), and hence it is natural to think of the Fredholm determinant Dix) of Kix\t, s),Q) Numbers in brackets refer to the references cited at the end of the paper.