2011
DOI: 10.1515/form.2011.046
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Invariant sets and ergodic decomposition of local semi-Dirichlet forms

Abstract: Weakly invariant set and strongly invariant setLet X be a separable metric space and m a σ-finite Borel measure on X. Let (T t ) t≥0 be a C 0 -semigroup on L 2 (X; m) and (T t ) t≥0 the dual C 0 -semigroup of (T t ) t≥0 on L 2 (X; m). An m-measurable subset B of X is said to be weakly invariant with respect to (T t ) t≥0 if I B c T t I B u = 0 for any t > 0 and u ∈ L 2 (X; m), equivalently B c is weakly invariant with respect to (T t ) t≥0 . An m-measurable subset B of X is said to be (strongly) invariant with… Show more

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Cited by 11 publications
(5 citation statements)
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“…The author is grateful to Professors Sergio Albeverio and Andreas Eberle, and to Dr. Kohei Suzuki, for fruitful conversations on the subject of the present work, and for respectively pointing out the references [1,13], and [3,20]. Finally, he is especially grateful to an anonymous Reviewer for their very careful reading and their suggestions which improved the readability of the paper.…”
Section: Acknowledgementsmentioning
confidence: 83%
“…The author is grateful to Professors Sergio Albeverio and Andreas Eberle, and to Dr. Kohei Suzuki, for fruitful conversations on the subject of the present work, and for respectively pointing out the references [1,13], and [3,20]. Finally, he is especially grateful to an anonymous Reviewer for their very careful reading and their suggestions which improved the readability of the paper.…”
Section: Acknowledgementsmentioning
confidence: 83%
“…Since in our case the associated generator may be non-symmetric and non-sectorial, the above results dealing with symmetric Dirichlet form theory may not apply. Therefore, we use the stronger concept of strict irreducibility of (T t ) t>0 covered in [23] and originally due to [36]. In [23,Section 3.2.3], under the assumption that µ is a Muckenhoupt A β -weight, β ∈ [1,2], and that (T t ) t>0 is associated to a symmetric Dirichlet form defined as the closure of (2.57), the pointwise lower bound of the associated heat kernel leads to the strict irreducibility of (T t ) t>0 .…”
Section: Lemma 318 Under the Assumption (A) Of Section 221 There Exis...mentioning
confidence: 99%
“…Remark 4.9. (a) We shall call E cons the conservative part of E, while E diss is the dissipative part of E. Besides, we shall name X cons , respectively X diss , the conservative, respectively the dissipative, space of E. Let us emphasize that our connotations for conservative and dissipative spaces differ from those introduced in [FOT11, p.55] or [Kuw11].…”
Section: Hence For Everymentioning
confidence: 99%