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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.
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