Dr. Moritz Gerlach

ehemaliger Mitarbeiter

Kontakt


Sprechzeiten:

nach Vereinbarung

...
  • Asymptotik von Halbgruppen und Markovprozessen
  • Ergodentheorie
  • Kernoperatoren, Positivität und Verbandstheorie

Journal articles

2018 | Convergence of Dynamics and the Perron-Frobenius Operator | Moritz Gerlach Zeitschrift: Israel Journal of Mathematics Seiten: 451–463 Band: 225(1) Link zur Publikation, Link zum Preprint

Convergence of Dynamics and the Perron-Frobenius Operator

Autoren: Moritz Gerlach (2018)

We complete the picture how the asymptotic behavior of a dynamical system is reflected by properties of the associated Perron-Frobenius operator. Our main result states that strong convergence of the powers of the Perron-Frobenius operator is equivalent to setwise convergence of the underlying dynamic in the measure algebra. This situation is furthermore characterized by a uniform mixing-like property of the system.

Zeitschrift:
Israel Journal of Mathematics
Seiten:
451–463
Band:
225(1)

2018 | Lower Bounds and the Asymptotic Behaviour of Positive Operator Semigroups | Moritz Gerlach, Jochen Glück Zeitschrift: Ergodic Theory and Dynamical Systems Seiten: 3012-3041 Band: 38(8) Link zur Publikation, Link zum Preprint

Lower Bounds and the Asymptotic Behaviour of Positive Operator Semigroups

Autoren: Moritz Gerlach, Jochen Glück (2018)

If $(T_t)$ 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 $t \to \infty$. 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.

Zeitschrift:
Ergodic Theory and Dynamical Systems
Seiten:
3012-3041
Band:
38(8)

2018 | Error estimates for spectral convergence of the graph Laplacian on random geometric graphs towards the Laplace--Beltrami operator | Nicolas Garcia Trillos, Moritz Gerlach, Matthias Hein, Dejan Slepcev Link zum Preprint

Error estimates for spectral convergence of the graph Laplacian on random geometric graphs towards the Laplace--Beltrami operator

Autoren: Nicolas Garcia Trillos, Moritz Gerlach, Matthias Hein, Dejan Slepcev (2018)

2017 | On a Convergence Theorem for Semigroups of Positive Integral Operators | Moritz Gerlach, Jochen Glück Zeitschrift: Comptes rendus - Mathematics Seiten: 973-976 Band: 355(9) Link zur Publikation, Link zum Preprint

On a Convergence Theorem for Semigroups of Positive Integral Operators

Autoren: Moritz Gerlach, Jochen Glück (2017)

We give a new and very short proof of a theorem of Greiner asserting that a positive and contractive c_0-semigroup on an L^p-space is strongly convergent in case that it has a strictly positive fixed point and contains an integral operator. Our proof is a streamlined version of a much more general approach to the asymptotic theory of positive semigroups developed recently by the authors. Under the assumptions of Greiner's theorem, this approach becomes particularly elegant and simple. We also give an outlook on several generalisations of this result.

Zeitschrift:
Comptes rendus - Mathematics
Seiten:
973-976
Band:
355(9)

2017 | Mean ergodicity vs weak almost periodicity | Moritz Gerlach, Jochen Glück Zeitschrift: Studia Mathematica, to appear Link zum Preprint

Mean ergodicity vs weak almost periodicity

Autoren: Moritz Gerlach, Jochen Glück (2017)

We provide explicit examples of positive and power-bounded operators on $c_0$ and $\ell^\infty$ which are mean ergodic but not weakly almost periodic. As a consequence we prove that a countably order complete Banach lattice on which every positive and power-bounded mean ergodic operator is weakly almost periodic is necessarily a KB-space. This answers several open questions from the literature.
Finally, we prove that if $T$ is a positive mean ergodic operator with zero fixed space on an arbitrary Banach lattice, then so is every power of $T$.

Zeitschrift:
Studia Mathematica, to appear

2017 | Convergence of Positive Operator Semigroups | Moritz Gerlach, Jochen Glück Link zum Preprint

Convergence of Positive Operator Semigroups

Autoren: Moritz Gerlach, Jochen Glück (2017)

We present new conditions for semigroups of positive operators to converge strongly as time tends to infinity. Our proofs are based on a novel approach combining the well-known splitting theorem by Jacobs, de Leeuw and Glicksberg with a purely algebraic result about positive group representations. Thus we obtain convergence theorems not only for one-parameter semigroups but for a much larger class of semigroup representations.
Our results allow for a unified treatment of various theorems from the literature that, under technical assumptions, a bounded positive $C_0$-semigroup containing or dominating a kernel operator converges strongly as $t \to \infty$. We gain new insights into the structure theoretical background of those theorems and generalise them in several respects; especially we drop any kind of continuity or regularity assumption with respect
to the time parameter.
As applications we derive, inter alia, a generalisation of a famous theorem by Doob for operator semigroups on the space of measures and a Tauberian theorem for positive one-parameter semigroups under rather weak continuity assumptions. We also demonstrate how our results are useful to treat semigroups that do not satisfy any irreducibility conditions.

2015 | On the lattice structure of kernel operators | Moritz Gerlach, Markus Kunze Zeitschrift: Mathematische Nachrichten Seiten: 584-592 Band: 288 Link zur Publikation, Link zum Preprint

On the lattice structure of kernel operators

Autoren: Moritz Gerlach, Markus Kunze (2015)

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.

Zeitschrift:
Mathematische Nachrichten
Seiten:
584-592
Band:
288

2014 | A Tauberian theorem for strong Feller semigroups | Moritz Gerlach Zeitschrift: Archiv der Mathematik Seiten: 245-255 Band: 3 Link zur Publikation, Link zum Preprint

A Tauberian theorem for strong Feller semigroups

Autoren: Moritz Gerlach (2014)

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.

Zeitschrift:
Archiv der Mathematik
Seiten:
245-255
Band:
3

2014 | Mean ergodic theorems on norming dual pairs | Moritz Gerlach, Markus Kunze Zeitschrift: Ergodic Theory and Dynamical Systems Seiten: 1210–1229 Band: 34 Link zur Publikation, Link zum Preprint

Mean ergodic theorems on norming dual pairs

Autoren: Moritz Gerlach, Markus Kunze (2014)

We extend the classical mean ergodic theorem to the setting of norming dual pairs. It turns out that, in general, not all equivalences from the Banach space setting remain valid in our situation. However, for Markovian semigroups on the norming dual pair (C_b(E), M(E)) all classical equivalences hold true under an additional assumption which is slightly weaker than the e-property.

Zeitschrift:
Ergodic Theory and Dynamical Systems
Seiten:
1210–1229
Band:
34

2013 | On the peripheral point spectrum and the asymptotic behavior of irreducible semigroups of Harris operators | Moritz Gerlach Zeitschrift: Positivity Seiten: 875–898 Band: 17 Link zur Publikation, Link zum Preprint

On the peripheral point spectrum and the asymptotic behavior of irreducible semigroups of Harris operators

Autoren: Moritz Gerlach (2013)

Given a positive, irreducible and bounded C_0-semigroup on a Banach lattice with order continuous norm, we prove that the peripheral point spectrum of its generator is trivial whenever one of its operators dominates a non-trivial compact or kernel operator. For a discrete semigroup, i.e. for powers of a single operator T, we show that the point spectrum of some power T^k intersects the unit circle at most in 1. As a consequence, we obtain a sufficient condition for strong convergence of the C_0-semigroup and for a subsequence of the powers of T, respectively.

Zeitschrift:
Positivity
Seiten:
875–898
Band:
17

2012 | A new proof of Doob's theorem | Moritz Gerlach, Robin Nittka Zeitschrift: Journal of Mathematical Analysis and Applications Seiten: 763–774 Band: 388 Link zur Publikation, Link zum Preprint

A new proof of Doob's theorem

Autoren: Moritz Gerlach, Robin Nittka (2012)

We prove that every bounded, positive, irreducible, stochastically continuous semigroup on the space of bounded, measurable functions which is strong Feller, consists of kernel operators and possesses an invariant measure converges pointwise. This differs from Doob's theorem in that we do not require the semigroup to be Markovian and request a fairly weak kind of irreducibility. In addition, we elaborate on the various notions of kernel operators in this context, show the stronger result that the adjoint semigroup converges strongly and discuss as an example diffusion equations on rough domains. The proofs are based on the theory of positive semigroups and do not use probability theory.

Zeitschrift:
Journal of Mathematical Analysis and Applications
Seiten:
763–774
Band:
388
  • M. Gerlach
    Semigroups of Kernel Operators
    PhD thesis (2014)

  • M. Gerlach
    The asymptotic behavior of positive semigroups
    diploma thesis (2010)

  • M. Gerlach
    Vergleich von Zeit- und Platzkomplexität
    diploma thesis (2008)