On error operators related to the arbitrary functions principle

Nicolas Bouleau (1)

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

The error on a real quantity Y due to the graduation of the measuring instrument may be asymptotically represented, when the graduation is regular and fines down, by a Dirichlet form on R whose square field operator does not depend on the probability law of Y as soon as this law possesses a continuous density. This feature is related to the "arbitrary functions principle" (Poincar'e, Hopf ). We give extensions of this property to Rd and to the Wiener space for some approximations of the Brownian motion. This gives new approximations of the Ornstein-Uhlenbeck gradient. These results apply to the discretization of some stochastic differential equations encountered in mechanics.

Mots clés: arbitrary functions; Dirichlet forms; Euler scheme; Girsanov theorem; mechanical system; Rajchman measure; square field operator; stable convergence; stochastic differential equation

Domaines: Probabilités [math.PR]

Nicolas Bouleau. On error operators related to the arbitrary functions principle. Journal of Functional Analysis, 2007, 251, pp.325-345. ⟨hal-00781408⟩