  We consider the Boussinesq approximation for Rayleigh-B\'{e}nard convection
perturbed by an additive noise and with boundary conditions corresponding to
heating from below. In two space dimensions, with sufficient stochastic forcing
in the temperature component and large Prandtl number, we establish the
existence of a unique ergodic invariant measure. In three space dimensions, we
prove the existence of a statistically invariant state, and establish unique
ergodicity for the infinite Prandtl Boussinesq system. Throughout this work we
provide streamlined proofs of unique ergodicity which invoke an asymptotic
coupling argument, a delicate usage of the maximum principle, and exponential
martingale inequalities. Lastly, we show that the background method of
Constantin-Doering [CD96] can be applied in our stochastic setting, and prove
bounds on the Nusselt number relative to the unique invariant measure.
