On error operators related to the arbitrary functions principle

TL;DR

This paper explores how measurement errors in real quantities can be modeled using Dirichlet forms, revealing invariance properties related to the arbitrary functions principle and extending these ideas to stochastic processes and differential equations.

Contribution

It extends the arbitrary functions principle to Dirichlet forms on R^d and Wiener space, providing new insights into error modeling and stochastic differential equation discretization.

Findings

Error operators can be represented by Dirichlet forms independent of the law of Y with a continuous density.

Extensions of the arbitrary functions principle to higher dimensions and Wiener space are established.

New approximation methods for the Ornstein-Uhlenbeck gradient are introduced.

Abstract

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.