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For a wide range of values of the incoming solar radiation, the Earthfeatures at least two attracting states, which correspond to competingclimates. The warm climate is analogous to the present one; the snowballclimate features global glaciation and conditions that can hardly support lifeforms. Paleoclimatic evidences suggest that in the past our planet flippedbetween these two states. The main physical mechanism responsible for such aninstability is the ice-albedo feedback. In a previous work, we defined theMelancholia states that sit between the two climates. Such states are embeddedin the boundaries between the two basins of attraction and feature extensiveglaciation down to relatively low latitudes. Here, we explore the globalstability properties of the system by introducing random perturbations asmodulations to the intensity of the incoming solar radiation. We observenoise-induced transitions between the competing basins of attraction. In theweak noise limit, large deviation laws define the invariant measure, thestatistics of escape times, and typical escape paths called instantons. Byconstructing the instantons empirically, we show that the Melancholia statesare the gateways for the noise-induced transitions. In the region ofmultistability, in the zero-noise limit, the measure is supported only on oneof the competing attractors. For low (high) values of the solar irradiance, thelimit measure is the snowball (warm) climate. The changeover between the tworegimes corresponds to a first-order phase transition in the system. Theframework we propose seems of general relevance for the study of complexmultistable systems. Finally, we put forward a new method for constructingMelancholia states from direct numerical simulations, black which provides apossible alternative with respect to the edge-tracking algorithm.

Global Stability Properties of the Climate: Melancholia States, Invariant Measures, and Phase Transitions