Covariance Inequalities Induced by Biorthogonal Operators

✍ Scribed by Eduardo Mayer-Wolf

• Book ID: 102972309
• Publisher: Elsevier Science
• Year: 1996
• Tongue: English
• Weight: 1003 KB
• Volume: 140
• Category: Article
• ISSN: 0022-1236
• DOI: 10.1006/jfan.1996.0104

✦ Synopsis

The logarithmic derivative of a weakly second order probability measure + on a general vector space X is an example of an operator from X into L 2 (+) which preserves the pairing of X and X$ under the inner product of L 2 (+). This biorthogonality property is exploited to obtain, by means of comparison, a lower bound on +'s covariance operator. The logarithmic derivative is then explicitly evaluated for probability measures induced by the solutions to stochastic ordinary and partial differential equations.

1996 Academic Press, Inc.

which itself is easily verified integrating by parts.

One consequence of (1.2) for V is that for a given Z and Gaussian G,