Variance reduction for diffusions

The most common way to sample from a probability distribution is to use Markov Chain Monte Carlo methods. One can find many diffusions with the target distribution as equilibrium measure, so that the state of the diffusion after a long time provides a good sample from that distribution. One naturally wants to choose the best algorithm. One way to do this is to consider a reversible diffusion, and add to it an antisymmetric drift which preserves the invariant measure. We prove that, in general, adding an antisymmetric drift reduces the asymptotic variance, and provide some extensions of this result.