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We prove stochastic stability of the three-dimensional Rayleigh-B\'enardconvection in the infinite Prandtl number regime for any pair of temperaturesmaintained on the top and the bottom. Assuming that the non-degenerate randomperturbation acts in a thin layer adjacent to the bottom of the domain, weprove that the random flow periodic in the two infinite directions stabilisesto a unique stationary measure, provided that there is at least one pointaccessible from any initial state. We also prove that the latter property issatisfied if the amplitude of the noise is sufficiently large.

Rayleigh-B\'enard convection with stochastic forcing localised near the bottom