Bismut's Way of the Malliavin Calculus for Non-Markovian Semi-groups: An introduction

The Jain-Monrad criterion for rough paths and applications to random Fourier series and non-Markovian Hörmander theory

We discuss stochastic calculus for large classes of Gaussian processes, based on rough path analysis. Our key condition is a covariance measure structure combined 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 of explicit expressions for the covariance or Volterra kernels. Of special interest are random Fourier series, with covariance given as Fourier series itself, and we formulate conditions directly in terms of the Fourier coefficients. We also establish convergence and rates of convergence in ro…