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We discuss stochastic calculus for large classes of Gaussian processes, basedon rough path analysis. Our key condition is a covariance measure structurecombined with a classical criterion due to Jain and Monrad [Ann. Probab. 11(1983) 46-57]. This condition is verified in many examples, even in absence ofexplicit expressions for the covariance or Volterra kernels. Of specialinterest are random Fourier series, with covariance given as Fourier seriesitself, and we formulate conditions directly in terms of the Fouriercoefficients. We also establish convergence and rates of convergence in roughpath metrics of approximations to such random Fourier series. An application toSPDE is given. Our criterion also leads to an embedding result forCameron-Martin paths and complementary Young regularity (CYR) of theCameron-Martin space and Gaussian sample paths. CYR is known to imply Malliavinregularity and also It\^o-like probabilistic estimates for stochastic integrals(resp., stochastic differential equations) despite their (rough) pathwiseconstruction. At last, we give an application in the context of non-MarkovianH\"ormander theory.

The Jain-Monrad criterion for rough paths and applications to random Fourier series and non-Markovian H\"ormander theory