On Gaussian measures equivalent to Wiener measure
暂无分享,去创建一个
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
[1] D. A. Woodward. A general class of linear transformations of Wiener integrals , 1961 .
[2] G. Baxter,et al. A strong limit theorem for Gaussian processes , 1956 .
[3] J. Doob. Stochastic processes , 1953 .
[4] Dale E. Varberg,et al. On equivalence of Gaussian measures , 1961 .
[5] Aaas News,et al. Book Reviews , 1893, Buffalo Medical and Surgical Journal.