Properties of convergence in Dirichlet structures

In univariate settings, we prove a strong reinforcement of the energy image density criterion for local Dirichlet forms admitting square field operators. This criterion enables us to redemonstrate some classical results of Dirichlet forms theory (Ancona, 1976 [2]). Besides, when X=(X1,…,Xp) belongs to the D domain of the Dirichlet form, and when its square field operator matrix Γ[X,tX] is almost surely definite, we prove that LX is a Rajchman measure. This is the first result in full generality in the direction of Bouleau–Hirsch conjecture. Moreover, in multivariate settings, we study the particular case of Sobolev spaces: we prove that a convergence for the Sobolev norm W1,p(Rd,Rp) toward a non-degenerate limit, entails convergence of push-forward measures in the total variation topology. Our proofs are based on a new kind of integration by parts which is of independent interest.