On Gaussian measures equivalent to Wiener measure

Introduction. The question of when two Gaussian measures are equivalent continues to receive attention (see, for example, [4]). In the present paper we nvestigate this question in the case where one of the measures is Wiener measure. We succeed in giving necessary and sufficient conditions for a wide class of Gaussian measures to be equivalent to Wiener measure (Theorem 5). Moreover, in the case of equivalence, we give an explicit formula for the Radon-Nikodym derivative of one measure with respect to the other. These results, which generalize earlier ones of the author [6], are based on a recent paper of Woodward [7] on linear transformations of the Wiener process. In particular, our sufficiency results are obtained by examining the question of when it is possible to represent a Gaussian process by means of a linear transformation of the Wiener process. In answering this question we are led to study a new class of Gaussian processes, namely those which have what we shall call factorable covariance functions. The covariance function r is factorable on [0, fc] if r may be written in the form