  We show that Langevin$-$Smoluchowski measure on path space is invariant under
time-reversal, followed by stochastic control of the drift with a novel
entropic-type criterion. Repeated application of these forward-backward steps
leads to a sequence of stochastic control problems, whose initial/terminal
distributions converge to the Gibbs probability measure of the diffusion, and
whose values decrease to zero along the relative entropy of the
Langevin$-$Smoluchowski flow with respect to this Gibbs measure.
