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 classical results of Dirichlet forms theory \cite{ancona1976continuité}. Besides, when $X=(X_1,\dots,X_p)$ belongs to the $\D$ domain of the Dirichlet form, and when its square field operator matrix $\Gamma[X,{}^t X]$ is almost surely definite, we prove that $\mathcal{L}_X$ is Rajchman. 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 show that a convergence for the Sobolev norm $\mathcal{W}^{1,p}(\R^d,\R^p)$ toward a non-degenerate limit entails convergence of push-forward measures in the total variation topology. \cite{bouleau1986formes}