When and how an error yields a Dirichlet form

We consider a random variable Y and approximations Y_n, , defined on the same probability space with values in the same measurable space as Y. We are interested in situations where the approximations Y_n allow to define a Dirichlet form in the space where is the law of Y. Our approach consists in studying both biases and variances. The article attempts to propose a general theoretical framework. It is illustrated by several examples.