Some historical aspects of error calculus by Dirichlet forms

We indicate some possible future research

Nicolas Bouleau


Scholarcy highlights

  • There are several kinds of error calculations which have not followed the same historical development
  • The error calculus by Dirichlet forms that we will explain and trace the origins has to be distinguished from the following calculations: a) The calculus of roundoff errors in numerical computations which appeared far before the representation of numbers in floating point be implemented on computers, and which possesses its specific difficulties
  • 1 n n i=1 xi is the best value to take in account, he showed, with some additional assumptions, that the errors follow necessarly a normal law and the arithmetic average is both the most likely value and the one given by the least squares method. Gauss tackled this question in the following way. First he admits – and this idea will be kept in the error theory with Dirichlet forms – that the quantity to be measured is random
  • It is in his Theoria Combinationis Observatonum Erroribus Minimis Obnoxiae published in 1823 that Gauss details his ideas about the errors propagation. He cites Laplace and discusses the merits of reasonning with repeated observations or with observations immediately erroneous. Behind this discussion is the fact that Laplace gave the first analytical proof of the central limit theorem, and that Gauss intends to assert the interests of his demonstration that if the arithmetic mean is taken as the correct value the law is necessarily normal, that he replaces in a more general approach of some kind of error calculus in an extended meaning
  • The main benefit of the extension tool is that error theory based on Dirichlet forms extends to the infinite dimension, which allows error calculus on stochastic processes
  • In the setting of Dirichlet forms we know that the carré du champ operator represents the variance of the error and that the bias of the error can be represented by the symmetric generator
  • We must get used to consider that any input process is accompanied with some accuracy defined by an error structure conditioning – depending on the stochasticity and the intrinsic accuracy of the treatment – the precision on the output expressed by an error structure, so that a new treatment may be applied

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