ct 2 00 6 Differential Calculus for Dirichlet forms : The measure - valued gradient preserved by image ∗ Nicolas BOULEAU Ecole

In order to develop a differential calculus for error propagation (cf [3]) 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. 1 Preamble Our main purpose being to study images, in order to avoid unessential difficulties, we restrict us to Dirichlet forms defined on probability spaces. On a probability space (W, W, m) let us consider a local Dirichlet form (D, E) ∗This work was presented at the meeting on Stochastic Analysis and Potential Theory, St Priest de Gimel, 1-6 sept 2002