The Entropy Distance between the Wiener and Stationary Gaussian Measures

Investigating the entropy distance between the Wiener measure,Wt0,τ , and stationary Gaussian measures, Qt0,τ on the space of continuous functions C[t0 − τ, t0 + τ ], we show that in some cases this distance can essentially be computed. This is done by explicitly computing a related quantity which in effect is a valid approximation of the entropy distance, provided it is sufficiently small; this will be the case if τ/t0 is small. We prove that H(Wt0,τ , Qt0,τ ) > τ/2t0, and then show that τ/2t0 is essentially the typical case of such entropy distance, provided the mean and the variance of the stationary measures are set “appropriately”. Utilizing a similar technique, we estimate the entropy distance between the Ornstein-Uhlenbeck measure and other stationary Gaussian measures on C[1− τ, 1 + τ ]. Using this result combined with a variant of the triangle inequality for the entropy distance, which we devise, yields an upper bound on the entropy distance between stationary Gaussian measures which are absolutely continuous with respect to the Wiener measure.