Convergence to the time average by stochastic regularization

In Ergodic Theory it is natural to consider the pointwise convergence of finite time averages of functions with respect to the flow of dynamical systems. Since the pointwise convergence is too weak for applications to Hamiltonian Perturbation Theory, requiring differentiability, we first introduce regularized averages obtained through a stochastic perturbation of an integrable Hamiltonian flow, and then we provide detailed estimates. In particular, for a special vanishing limit of the stochastic perturbation, we obtain convergence even in a Sobolev norm taking into account the derivatives.