Diffusion with nonlocal Dirichlet boundary conditions on domains

Kunze, Markus

doi:10.4064/sm181012-24-5

Cited by 8 publications

(8 citation statements)

References 35 publications

(51 reference statements)

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“…However, there are important instances of operator semigroups which lack strong continuity, e.g., the heat semigroup on the space of bounded continuous functions on R or on the space of finite measures over R or, in an abstract context, dual semigroups of C 0 -semigroups on non-reflexive spaces. (See the recent paper [Kun18] for a more involved concrete example.) Hence, there is a need for results on asymptotics beyond C 0 -semigroups.…”

Section: Introductionmentioning

confidence: 99%

Asymptotics of Operator Semigroups via the Semigroup at Infinity

Glück

Haase

2019

Trends in Mathematics

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Dedicated to Ben de Pagter on the occasion of his 65th birthday.Abstract. We systematize and generalize recent results of Gerlach and Glück on the strong convergence and spectral theory of bounded (positive) operator semigroups (Ts) s∈S on Banach spaces (lattices). (Here, S can be an arbitrary commutative semigroup, and no topological assumptions neither on S nor on its representation are required.) To this aim, we introduce the "semigroup at infinity" and give useful criteria ensuring that the well-known Jacobs-de Leeuw-Glicksberg splitting theory can be applied to it.Next, we confine these abstract results to positive semigroups on Banach lattices with a quasi-interior point. In that situation, the said criteria are intimately linked to so-called AM-compact operators (which entail kernel operators and compact operators); and they imply that the original semigroup asymptotically embeds into a compact group of positive invertible operators on an atomic Banach lattice. By means of a structure theorem for such group representations (reminiscent of the Peter-Weyl theorem and its consequences for Banach space representations of compact groups) we are able to establish quite general conditions implying the strong convergence of the original semigroup.Finally, we show how some classical results of Greiner (1982), Davies (2005), Keicher (2006) and Arendt (2008) and more recent ones by Gerlach and Glück (2017) are covered and extended through our approach.

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Section: Introductionmentioning

confidence: 99%

Asymptotics of Operator Semigroups via the Semigroup at Infinity

Glück

Haase

2019

Trends in Mathematics

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“…As a consequence of the (classical) maximum principle, we see that the semigroups T n are increasing in n, thus T is the supremum of the semigroups T n . Taking Laplace transforms, we see that also the resolvents are increasing and the supremum of the resolvents is the resolvent of the generator of T , see [36,Proposition 2.12] for an appropriate version of this result. Making use of the Lyapunov function V again, one can identify the operator A as the generator of T .…”

Section: Sketch Of Proofmentioning

confidence: 99%

“…There are, however, also examples of transition semigroups which are not even pointwise continuous. This is for example the case when considering nonlocal boundary conditions see [6,36]. For a general treatment of time continuity properties of Markov semigroups on spaces of measures we refer to [28].…”

Section: Introductionmentioning

confidence: 99%

Stability of transition semigroups and applications to parabolic equations

Gerlach¹,

Glück²,

Kunze³

2020

Preprint

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We study the long-term behavior of positive operator semigroups on spaces of bounded functions and on spaces of signed measures; such semigroups frequently appear in the study of parabolic partial differential equations and in stochastic analysis. One of our main results is a Tauberian type theorem which characterizes the convergence of a strongly Feller semigroup -uniformly on compact sets or in total variation norm -in terms of an ergodicity condition for the fixed spaces of the semigroup and its dual. In addition, we present a generalization of a classical convergence theorem of Doob to semigroups with arbitrarily large fixed space.Throughout, we require only very weak assumptions on the involved semigroups; in particular, none of our convergence theorems requires any time regularity assumptions. This is possible due to recently developed methods in the asymptotic theory of semigroups on Banach lattices, which focus on algebraic rather than topological properties of the semigroup.Our results enable us to efficiently analyse the long-term behavior of various parabolic equations and systems, even if the coefficients of the corresponding differential operator are unbounded.

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“…The same references also show that the condition that Ω have Lipschitz boundary can be somewhat relaxed.One can also study non-local Robin boundary conditions instead of non-local Dirichlet boundary conditions; see [37, Section 4] and [39]. Similar questions as in Example 4.1 can also be studied on unbounded domains; this is the content of the recent article [40]. …”

Section: Applications Imentioning

confidence: 99%

“…Similar questions as in Example 4.1 can also be studied on unbounded domains; this is the content of the recent article [40].…”

Section: Applications Imentioning

confidence: 99%

Positive irreducible semigroups and their long-time behaviour

Arendt

Glück

2020

Phil. Trans. R. Soc. A.

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The notion Perron–Frobenius theory usually refers to the interaction between three properties of operator semigroups: positivity, spectrum and long-time behaviour. These interactions gives rise to a profound theory with plenty of applications. By a brief walk-through of the field and with many examples, we highlight two aspects of the subject, both related to the long-time behaviour of semigroups: (i) The classical question how positivity of a semigroup can be used to prove convergence to an equilibrium as t → ∞. (ii) The more recent phenomenon that positivity itself sometimes occurs only for large t , while being absent for smaller times. This article is part of the theme issue ‘Semigroup applications everywhere’.

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Diffusion with nonlocal Dirichlet boundary conditions on domains

Cited by 8 publications

References 35 publications

Asymptotics of Operator Semigroups via the Semigroup at Infinity

Asymptotics of Operator Semigroups via the Semigroup at Infinity

Stability of transition semigroups and applications to parabolic equations

Positive irreducible semigroups and their long-time behaviour

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