Almost periodicity of stochastic operators on $\ell^1(\mathbb{N})$

Keicher, Vera

doi:10.32513/tbilisi/1528768826

Cited by 3 publications

(3 citation statements)

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“…Since (T (t)| P X ) t≥0 is periodic, one has 1 ∈ σ p (B| T ). Hence, [20,Proposition 2.1] shows that there exists a strictly positive fixed vector x ∈ 1 of B and this fixed vector is clearly an element of P 1 . In particular, f ⊗ x is an element of…”

Section: Theorem 46mentioning

confidence: 99%

On the asymptotic behaviour of semigroups for flows in infinite networks

Alexander

2022

Semigroup Forum

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We study transport processes on infinite networks. The solution of these processes can be modeled by an operator semigroup on a suitable Banach space. Classically, such semigroups are strongly continuous and therefore their asymptotic behaviour is quite well understood. However, recently new examples of transport processes emerged where the corresponding semigroup is not strongly continuous. Due to this lack of strong continuity, there are currently only few results on the long-term behaviour of these semigroups. In this paper, we discuss the asymptotic behaviour for a certain class of these transport processes. In particular, it is proved that the solution semigroups behave asymptotically periodic with respect to the operator norm as a consequence of a more general result on the long-term behaviour by positive semigroups containing a multiplication operator. Furthermore, we revisit known results on the asymptotic behaviour of transport processes on infinite networks and prove the asymptotic periodicity of their extensions to the space of bounded measures.

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Section: Theorem 46mentioning

confidence: 99%

On the asymptotic behaviour of semigroups for flows in infinite networks

Alexander

2022

Semigroup Forum

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show abstract

“…Since (T (t)| P X ) t≥0 is periodic, one has 1 ∈ σ p (B| T ). Hence, [21,Proposition 2.1] shows that there exists a strictly positive fixed vector x ∈ ℓ 1 of B and this fixed vector is clearly an element of Pℓ 1 . In particular,…”

Section: 2mentioning

confidence: 99%

On the asymptotic behaviour of semigroups for flows in infinite networks

Alexander¹

2020

Preprint

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We study transport processes on infinite networks which can be modeled by an operator semigroup on a Banach space. Classically, such semigroups are strongly continuous and therefore their asymptotic behaviour is quite well understood. However, recently new examples of transport processes emerged in which the corresponding semigroup is not strongly continuous. Due to this lack of strong continuity, there are currently no results on the long-term behaviour of these semigroups. In this paper, we close this gap for a certain class of transport processes. In particular, it is proven that the solution semigroups behave asymptotically periodic with respect to the operator norm as a consequence of a more general result on the long-term behaviour by positive semigroups that contain a multiplication operator. Furthermore, we revisit known results on the asymptotic behaviour of transport processes on infinite networks and prove the asymptotically periodicity of the extensions of those semigroups to the space of bounded measures.

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“…The basis for our further investigation is the characterization due to F.G. Foster known as Proof (i) ⇔ (ii) is Foster's Theorem (for a functional analytic proof, more in the line of this paper, see [19,Thm. 2.11]), while (ii) ⇒ (iii) is the assertion of our Theorem 1 and the converse (iii) ⇒ (ii) is the necessary condition Lemma 9.…”

Section: Positive Recurrent Graphsmentioning

confidence: 99%