On some errors related to the graduation of measuring instruments

Abstract

The error on a real quantity Y due to the graduation of the measuring instrument may be represented, when the graduation is regular and fines down, by a Dirichlet form on R whose square field operator do 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" (Poincaré, Hopf). We give extensions of this property to multivariate case and infinite dimensional case for approximations of the Brownian motion. We use a Girsanov theorem for Dirichlet forms which has its own interest. Connections are given with discretization of stochastic differential equations.