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The detection of terrestrial planets by radial velocity and photometry ishindered by the presence of stellar signals. Those are often modeled asstationary Gaussian processes, whose kernels are based on qualitativeconsiderations, which do not fully leverage the existing physical understandingof stars. Our aim is to build a formalism which allows to transfer theknowledge of stellar activity into practical data analysis methods. Inparticular, we aim at obtaining kernels with physical parameters. This has twopurposes: better modelling signals of stellar origin to find smallerexoplanets, and extracting information about the star from the statisticalproperties of the data. We consider several observational channels such asphotometry, radial velocity, activity indicators, and build a model calledFENRIR to represent their stochastic variations due to stellar surfaceinhomogeneities. We compute analytically the covariance of this multi-channelstochastic process, and implement it in the S+LEAF framework to reduce the costof likelihood evaluations from $O(N^3)$ to $O(N)$. We also compute analyticallyhigher order cumulants of our FENRIR model, which quantify its non-Gaussianity.We obtain a fast Gaussian process framework with physical parameters, which weapply to the HARPS-N and SORCE observations of the Sun, and constrain a solarinclination compatible with the viewing geometry. We then discuss theapplication of our formalism to granulation. We exhibit non-Gaussianity insolar HARPS radial velocities, and argue that information is lost when stellaractivity signals are assumed to be Gaussian. We finally discuss the origin ofphase shifts between RVs and indicators, and how to build relevant activityindicators. We provide an open-source implementation of the FENRIR Gaussianprocess model with a Python interface.

A statistical model of stellar variability. I. FENRIR: a physics-based model of stellar activity, and its fast Gaussian process approximation