Covariance Inequalities Induced by Biorthogonal Operators

The logarithmic derivative of a weakly second order probability measureμon a general vector spaceXis an example of an operator fromXintoL2(μ) which preserves the pairing ofXandX′ under the inner product ofL2(μ). This biorthogonality property is exploited to obtain, by means of comparison, a lower bound onμʹs covariance operator. The logarithmic derivative is then explicitly evaluated for probability measures induced by the solutions to stochastic ordinary and partial differential equations.