Properties of convergence in Dirichlet structures

Abstract : 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}.
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Pré-publication, Document de travail
2012
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https://hal.archives-ouvertes.fr/hal-00691126
Contributeur : Guillaume Poly <>
Soumis le : mercredi 25 avril 2012 - 12:30:56
Dernière modification le : lundi 29 mai 2017 - 14:25:03
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  • HAL Id : hal-00691126, version 1

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Dominique Malicet, Guillaume Poly. Properties of convergence in Dirichlet structures. 2012. <hal-00691126>

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