Differential calculus for Dirichlet forms: The measure-valued gradient preserved by image

In order to develop a differential calculus for error propagation of Bouleau [Error Calculus for Finance and Physics, the Language of Dirichlet forms, De Gruyter, Berlin, 2003], we study local Dirichlet forms on probability spaces with carré du champ 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.