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We quantify the effect of Gaussian white noise on fast--slow dynamicalsystems with one fast and two slow variables, which display mixed-modeoscillations owing to the presence of a folded-node singularity. The stochasticsystem can be described by a continuous-space, discrete-time Markov chain,recording the returns of sample paths to a Poincar\'e section. We provideestimates on the kernel of this Markov chain, depending on the systemparameters and the noise intensity. These results yield predictions on theobserved random mixed-mode oscillation patterns. Our analysis shows that thereis an intricate interplay between the number of small-amplitude oscillationsand the global return mechanism. In combination with a local saturationphenomenon near the folded node, this interplay can modify the number ofsmall-amplitude oscillations after a large-amplitude oscillation. Finally,sufficient conditions are derived which determine when the noise increases thenumber of small-amplitude oscillations and when it decreases this number.

From random Poincar\'e maps to stochastic mixed-mode-oscillation patterns