The purpose of this work is to study some possible application of FKG inequality to the Brownian motion and to Stochastic Differential Equations. We introduce a special ordering on the Wiener space and prove the FKG inequality with respect to this ordering. Then we apply this result on the solutions $X_t$ of a stochastic differential equation with a positive coefficient $\sigma$, we prove that these solutions $X_t$ are increasing with respect to the ordering, and finally we deduce a correlation inequality between the solution of different stochastic equations.
"FKG Inequality for Brownian Motion and Stochastic Differential Equations." Electron. Commun. Probab. 10 7 - 16, 2005. https://doi.org/10.1214/ECP.v10-1127