  Gaussian Processes (GPs) are a versatile method that enables different
approaches towards learning for dynamics and control. Gaussianity assumptions
appear in two dimensions in GPs: The positive semi-definite kernel of the
underlying reproducing kernel Hilbert space is used to construct the
co-variance of a Gaussian distribution over functions, while measurement noise
(i.e. data corruption) is usually modeled as i.i.d. additive Gaussian. In this
note, we relax the latter Gaussianity assumption, i.e., we consider kernel
ridge regression with additive i.i.d. non-Gaussian measurement noise. To apply
the usual kernel trick, we rely on the representation of the uncertainty via
polynomial chaos expansions, which are series expansions for random variables
of finite variance introduced by Norbert Wiener. We derive and discuss the
analytic $\mathcal{L}^2$ solution to the arising Wiener kernel regression.
Considering a polynomial system as numerical example, we show that our approach
allows to untangle the effects of epistemic and aleatoric uncertainties.
