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We consider a class of dissipative stochastic differential equations (SDE's)with time-periodic coefficients in finite dimension, and the response oftime-asymptotic probability measures induced by such SDE's to sufficientlyregular, small perturbations of the underlying dynamics. Understanding such aresponse provides a systematic way to study changes of statistical observablesin response to perturbations, and it is often very useful for sensitivityanalysis, uncertainty quantification, and for improving probabilisticpredictions of nonlinear dynamical systems, especially in high dimensions.Here, we are concerned with the linear response to small perturbations in thecase when the time-asymptotic probability measures are time-periodic. First, weestablish sufficient conditions for the existence of stable randomtime-periodic orbits generated by the underlying SDE. Ergodicity oftime-periodic probability measures supported on these random periodic orbits issubsequently discussed. Then, we derive the so-called fluctuation-dissipationrelations which allow to describe the linear response of statisticalobservables to small perturbations away from the time-periodic ergodic regimein a manner which only exploits the unperturbed dynamics. The results areformulated in an abstract setting but they apply to problems ranging fromaspects of climate modelling, to molecular dynamics, to the study ofapproximation capacity of neural~networks and robustness of their estimates.

Time-periodic measures, random periodic orbits, and the linear response for dissipative non-autonomous stochastic differential equations