Autour de la variance comme forme de Dirichlet : filtrations et résolution de l'identité, contractions et BMO, espérances conditionnelles et principe complet du maximum

International audience ; We examine which consequences may be drawn from the simple fact that the variance of a random variable is a Dirichlet form. We obtain a caracterisation of self-adjoint operators whose resolution of identity comes from a family of conditional expectations. This also enables us to enlighten the fact that contractions act on BMO, and to prove that positive mixtures of conditional expectations satisfy the complète maximum principle.