Theoreme de Donsker et formes de Dirichlet

We use the language of errors to handle local Dirichlet forms with squared field operator (cf. [N. Bouleau, Error Calculus for Finance and Physics, the Language of Dirichlet Forms, De Gruyter, 2003]). Let us consider, under the hypotheses of Donsker theorem, a random walk converging weakly to a Brownian motion. If, in addition, the random walk is supposed to be erroneous, the convergence occurs in the sense of Dirichlet forms and induces the Ornstein-Uhlenbeck structure on the Wiener space. This quite natural result uses an extension of Donsker theorem to functions with quadratic growth. As an application we prove an invariance principle for the gradient of the maximum of the Brownian path computed by Nualart and Vives.