When and how an error yields a Dirichlet form

Nicolas Bouleau (1)

1 CERMICS - Centre d'Enseignement et de Recherche en Mathématiques, Informatique et Calcul Scientifique

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 $L^2(P_Y)$ where $P_Y$ 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.

Mots clés: error; approximation; Dirichlet form; square field operator; bias; Wiener space; stochastic differential equation

Domaines: Analyse fonctionnelle [math.FA]; Probabilités [math.PR]

Nicolas Bouleau. When and how an error yields a Dirichlet form. Journal of Functional Analysis, 2006, 240, pp.445-494. ⟨hal-00105511⟩