Some covariance inequalities in Wiener space

In this work we give an account of some covariance inequalities in abstract Wiener space. An FKG inequality is obtained with positivity and monotonicity being defined in terms of a given cone in the underlying Cameron–Martin space. The last part is dedicated to convex and log-concave functionals, including a proof of the Gaussian conjecture for a particular class of log-concave Wiener functionals.