Stochastic climate dynamics: Random attractors and time-dependent invariant measures

► Measure-theoretic concepts of random dynamical systems theory are introduced. ► Random SRB measures of nonlinear stochastic systems are then considered. ► Two idealized models of fundamental interest for climate dynamics serve as examples. ► The geometry of the corresponding random SRB measures is numerically explored. ► The physical insights provided by these mathematical objects are enlightened.