On the construction and convergence of traces of forms

BelHadjAli, Hichem; BenAmor, Ali; Seifert, Christian; Thabet, Amina

doi:10.1016/j.jfa.2019.05.017

Cited by 9 publications

(7 citation statements)

References 15 publications

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“…Proof. As for λ ≤ 0 the form E λ is a Dirichlet form, the first assertion follows from [BBST19] and the fact that Dirichlet forms have sub-Markovian semigroups.…”

Section: Asymptotic and Positivitymentioning

confidence: 95%

Spectral asymptotic and positivity for singular Dirichlet-to-Neumann operators

BenAmor¹

2020

Preprint

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In the framework of Hilbert spaces we shall give necessary and sufficient conditions to define a Dirichlet-to-Neumann operator via Dirichlet principle. For singular Dirichlet-to-Neumann operators we will establish Laurent expansion near singularities as well as Mittag-Leffler expansion for the related quadratic form. The established results will be exploited to solve definitively the problem of positivity of the related semigroup in the L 2 setting. The obtained results are supported by some examples on Lipschitz domains. Among other results, we shall demonstrate that regularity of the boundary may affect positivity and derive Mittag-Leffler expansion for the eigenvalues of singular Dirichlet-to-Neumann operators.

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“…Proof. As for λ ≤ 0 the form E λ is a Dirichlet form, the first assertion follows from [BBST19] and the fact that Dirichlet forms have sub-Markovian semigroups.…”

Section: Asymptotic and Positivitymentioning

confidence: 95%

Spectral asymptotic and positivity for singular Dirichlet-to-Neumann operators

BenAmor¹

2020

Preprint

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“…Proof. We follow the construction made in [BBST19]. For each λ > 0, we set P λ the E λ -orthogonal projection of D onto the E λ -orthogonal of ker J, ker J ⊥ E λ .…”

Section: Moreover If Y Is Invariant Then the Quadratic Form E Y Defin...mentioning

confidence: 99%

Decomposition formulae for Dirichlet forms and their corollaries

BenAmor¹,

Moussa²

2019

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We provide decompositions of Dirichlet forms into recurrent and transient parts as well as into conservative and dissipative parts, in the framework of Hausdorff state spaces. Combining both formulae we write every Dirichlet form as the sum of a recurrent, dissipative and transient conservative Dirichlet forms. Besides, we prove that Mosco convergence preserves invariant sets and that a Dirichlet form shares the same invariants sets with its approximating Dirichlet forms E (t) and E (β) . Finally we show the equivalence between conservativeness (resp. dissipativity) of a Dirichlet form and the conservativeness (reps. dissipativity) of E (t) and E (β) . The elaborated results are enlightened by some examples.

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“…Despite the increasing number of applications and its usage in various fields, schematic guides to deal with Mosco convergence and related results in a general setting are rather rare to find. With [14,15,6,28,23,27] we would like to name some sophisticated works, which fall into this category. Our motivation to derive and present the abstract theory in this survey is to provide a suitable groundwork in the field of Dirichlet forms to address problems form statistical mechanics.…”

Section: Introductionmentioning

confidence: 99%

Mosco convergence of gradient forms with non-convex interaction potential

Grothaus,

Wittmann

2021

Preprint

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Convexity of interaction potentials is a typical condition for the derivation of gradient bounds for diffusion semigroups in stochastic interface models, particle systems, etc. Gradient bounds are often used to show convergence of semigroups. However for a large class of convergence problems the assumption of convexity fails. The article suggests a way to overcome this hindrance, as it presents a new approach which is not based on gradient bounds. Combining the theory of Dirichlet forms with methods from numerical analysis we find abstract criteria for Mosco convergence of standard gradient forms with varying reference measures. These include cases in which the measures are not log-concave. To demonstrate the accessibility of our abstract theory we discuss a first application, generalizing an approximation result from [7], which first appeared in 2011.

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On the construction and convergence of traces of forms

Cited by 9 publications

References 15 publications

Spectral asymptotic and positivity for singular Dirichlet-to-Neumann operators

Spectral asymptotic and positivity for singular Dirichlet-to-Neumann operators

Decomposition formulae for Dirichlet forms and their corollaries

Mosco convergence of gradient forms with non-convex interaction potential

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