Differential calculus for Dirichlet forms : the measure-valued gradient preserved by image
Abstract
In order to develop a differential calculus for error propagation we study local Dirichlet forms on probability spaces with square field operator
Γ
-- i.e. error structures -- and we are looking for an object related to
Γ
which is linear and with a good behaviour by images. For this we introduce a new notion called the measure valued gradient which is a randomized square root of
Γ
. The exposition begins with inspecting some natural notions candidate to solve the problem before proposing the measure-valued gradient and proving its satisfactory properties.