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The entropy distance between the Wiener and stationary Gaussian measures

Keich, U. (1999) The entropy distance between the Wiener and stationary Gaussian measures. Pacific Journal of Mathematics, 188 (1). pp. 109-128. ISSN 0030-8730. http://resolver.caltech.edu/CaltechAUTHORS:KEIpjm99b

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Abstract

Investigating the entropy distance between the Wiener measure, W-t0,W- (tau) and stationary Gaussian measures, Q(t0, tau) on the space of continuous functions C[t(0) - tau, t(0) + tau], 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 tau/t(0) is small. We prove that H(Wt(0, tau), Q(t0, tau)) > tau/2(t0), and then show that tau/2t(0) 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 - tau, 1 + tau]. 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.


Item Type:Article
Additional Information:© Copyright 1999 Pacific Journal of Mathematics. Received July 6, 1998. This paper is based on my Ph.D. dissertation research under the supervision of Prof. H.P. McKean, Courant Institute, NYU. I am indebted to H. McKean, S.R.S. Varadhan and P. Deift for their helpful comments.
Record Number:CaltechAUTHORS:KEIpjm99b
Persistent URL:http://resolver.caltech.edu/CaltechAUTHORS:KEIpjm99b
Alternative URL:http://pjm.math.berkeley.edu/1999/188-1/p06.html
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:713
Collection:CaltechAUTHORS
Deposited By: Tony Diaz
Deposited On:16 Sep 2005
Last Modified:26 Dec 2012 08:41

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