A Tauberian theorem for strong Feller semigroups

Gerlach, Moritz

doi:10.1007/s00013-014-0619-3

Cited by 5 publications

(3 citation statements)

References 15 publications

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“…Our strategy is similar to the one in [11], however due to the abstract results of Sections 2 and 3 the proof simplifies as we can work directly within the lattice of kernel operators. This complements recent results about mean ergodicity of semigroups of kernel operators, see [10] and [9].…”

Section: Introductionsupporting

confidence: 88%

On the lattice structure of kernel operators

Gerlach

Kunze

2014

Math. Nachr.

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Abstract. Consider the lattice of bounded linear operators on the space of Borel measures on a Polish space. We prove that the operators which are continuous with respect to the weak topology induced by the bounded measurable functions form a sublattice that is lattice isomorphic to the space of transition kernels. As an application we present a purely analytic proof of Doob's theorem concerning stability of transition semigroups.

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Section: Introductionsupporting

confidence: 88%

On the lattice structure of kernel operators

Gerlach

Kunze

2014

Math. Nachr.

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“…[10, Thm VI.1.1] and [40,Thm 4.2]) while such criteria are more delicate in the infinite dimensional case, and in particular on non-discrete state spaces (see e.g. [6,Sec 4.2] and [20,Thm 4.4] for different versions and proofs of a classical theorem of Doob which addresses this issue; see also [19,Thm 3.6] for a related Tauberian theorem).…”

Section: Introductionmentioning

confidence: 99%

Lower bounds and the asymptotic behaviour of positive operator semigroups

Gerlach¹,

Glück²

2017

Ergod. Th. Dynam. Sys.

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If (Tt) is a semigroup of Markov operators on an L 1 -space that admits a non-trivial lower bound, then a well-known theorem of Lasota and Yorke asserts that the semigroup is strongly convergent as t → ∞. In this article we generalise and improve this result in several respects.First, we give a new and very simple proof for the fact that the same conclusion also holds if the semigroup is merely assumed to be bounded instead of Markov. As a main result we then prove a version of this theorem for semigroups which only admit certain individual lower bounds. Moreover, we generalise a theorem of Ding on semigroups of Frobenius-Perron operators. We also demonstrate how our results can be adapted to the setting of general Banach lattices and we give some counterexamples to show optimality of our results.Our methods combine some rather concrete estimates and approximation arguments with abstract functional analytical tools. One of these tools is a theorem which relates the convergence of a time-continuous operator semigroup to the convergence of embedded discrete semigroups.

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“…P f = Ω f dν ⋆ •1. It follows from [17,Corollary 3.7] (a related result can be found in Version 1 of [18] on the arxiv), that for every ν ∈ M (Ω) we have T ′ µ (t)ν → P ′ ν in total variation norm as t → ∞. From this it easily follows that T µ (t)f → P f with respect to σ(C b (Ω), M (Ω)) as t → ∞.…”

Section: Asymptotic Behaviormentioning

confidence: 99%

Diffusion with nonlocal Dirichlet boundary conditions on domains

Kunze

2020

Studia Math.

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We consider a second order differential operator A on an open and Dirichlet regular set Ω ⊂ R d (which typically is unbounded) and subject to nonlocal Dirichlet boundary conditions of the formHere, µ : ∂Ω → M (Ω) takes values in the probability measures on Ω and is continuous in the weak topoly σ(M (Ω), C b (Ω)). Under suitable assumptions on the coefficients in A , which may be unbounded, we prove that a realization Aµ of A subject to the nonlocal boundary condition, generates a (not strongly continuous) semigroup on L ∞ (Ω). We establish a sufficient condition for this semigroup to be Markovian and prove that in this case, it enjoys the strong Feller property. We also study the asymptotic behavior of the semigroup.2010 Mathematics Subject Classification. 47D07, 60J35, 35B40.

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A Tauberian theorem for strong Feller semigroups

Abstract: Abstract. We prove that a weakly ergodic, eventually strong Feller semigroup on the space of measures on a Polish space converges strongly to a projection onto its fixed space.

Cited by 5 publications

References 15 publications

On the lattice structure of kernel operators

On the lattice structure of kernel operators

Lower bounds and the asymptotic behaviour of positive operator semigroups

Diffusion with nonlocal Dirichlet boundary conditions on domains

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