On the Importance of the Levy Area for Studying the Limits of Functions of Converging Stochastic Processes. Application to Homogenization

Two concrete examples show us that the convergence of a family of stochastic processes "as controls", i.e. as integrators of SDEs or differential forms, may require more information than simply the limit in the uniform norm of the processes. This may be particularly important when one deals with the homogenization theory. The theory of rough paths is then used to bring some new results about interchanging limits and functionals of stochastic processes