  The concept of effective order is a popular methodology in the deterministic
literature for the construction of efficient and accurate integrators for
differential equations over long times. The idea is to enhance the accuracy of
a numerical method by using an appropriate change of variables called the
processor. We show that this technique can be extended to the stochastic
context for the construction of new high order integrators for the sampling of
the invariant measure of ergodic systems. The approach is illustrated with
modifications of the stochastic $\theta$-method applied to Brownian dynamics,
where postprocessors achieving order two are introduced. Numerical experiments,
including stiff ergodic systems, illustrate the efficiency and versatility of
the approach.
