A PDE approach to finite time approximations in Ergodic Theory

For dynamical systems defined by vector fields over a compact invariant set, we introduce a new class of approximated first integrals based on finite-time averages and satisfying an explicit first order partial differential equation. Referring to the PDE framework, we detect their viscosity robustness with respect to stochastic perturbations of the vector field. We formulate this approximating device also in the Lyapunov exponents framework. In such a case, we compare the operative use of the introduced approximated first integrals to the common use of the Fast Lyapunov Indicators to detect the phase space structure of quasi-integrable systems.