Transformations of Wiener integrals under a general class of linear transformations

Abstract

Introduction. Let C be the space of all real-valued functions x(t) continuous in 0 < t ?1, and vanishing at t = 0. Wiener has defined a measure over the space C and in terms of this measure he has defined an average or integral over C which is intimately related among other things to the theory of the Brownian motion [1, 2](1). The present authors have recently investigated certain aspects of the Wiener integral [3, 4] and have obtained for instance in [4] a result which shows how the integral is transformed under translations. Irn the present paper we determine how the integral transforms under a certain class of linear homogeneous transformations. This result is also combined with the earlier result on translations to yield a transformation formula for the nonhomogeneous transformation-translation plus linear homogeneous transformation. By applying the transformation formula to the special linear transformation