# Dr. Moritz Gerlach

### freier Mitarbeiter

Kontakt

#### Sprechzeiten:

nach Vereinbarung

#### Forschungsinteressen

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

#### Journal articles

2020 | Stability of transition semigroups and applications to parabolic equations | Moritz Gerlach, Jochen Glück, Markus KunzeZeitschrift: submittedLink zum Preprint

### Stability of transition semigroups and applications to parabolic equations

#### Autoren: Moritz Gerlach, Jochen Glück, Markus Kunze (2020)

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.

Zeitschrift:
submitted

2019 | Convergence of Positive Operator Semigroups | Moritz Gerlach, Jochen GlückZeitschrift: Transactions of the American Mathematical SocietySeiten: 6603-6627Band: 372Link zur Publikation , Link zum Preprint

### Convergence of Positive Operator Semigroups

#### Autoren: Moritz Gerlach, Jochen Glück (2019)

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.

Zeitschrift:
Transactions of the American Mathematical Society
Seiten:
6603-6627
Band:
372

2019 | Mean ergodicity vs weak almost periodicity | Moritz Gerlach, Jochen GlückZeitschrift: Studia MathematicaSeiten: 45-56Band: 248Link zur Publikation , Link zum Preprint

### Mean ergodicity vs weak almost periodicity

#### Autoren: Moritz Gerlach, Jochen Glück (2019)

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
Seiten:
45-56
Band:
248

2019 | 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 SlepcevZeitschrift: Foundations of Computational MathematicsLink zur Publikation , 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 (2019)

We study the convergence of the graph Laplacian of a random geometric graph generated by an i.i.d. sample from a -dimensional submanifold in as the sample size increases and the neighborhood size tends to zero. We show that eigenvalues and eigenvectors of the graph Laplacian converge with a rate of to the eigenvalues and eigenfunctions of the weighted Laplace-Beltrami operator of .
No information on the submanifold is needed in the construction of the graph or the "out-of-sample extension" of the eigenvectors. Of independent interest is a generalization of the rate of convergence of empirical measures on submanifolds in in infinity transportation distance.

Zeitschrift:
Foundations of Computational Mathematics

2018 | Convergence of Dynamics and the Perron-Frobenius Operator | Moritz GerlachZeitschrift: Israel Journal of MathematicsSeiten: 451–463Band: 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 GlückZeitschrift: Ergodic Theory and Dynamical SystemsSeiten: 3012-3041Band: 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)

2017 | On a Convergence Theorem for Semigroups of Positive Integral Operators | Moritz Gerlach, Jochen GlückZeitschrift: Comptes rendus - MathematicsSeiten: 973-976Band: 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)

2015 | On the lattice structure of kernel operators | Moritz Gerlach, Markus KunzeZeitschrift: Mathematische NachrichtenSeiten: 584-592Band: 288Link 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 GerlachZeitschrift: Archiv der MathematikSeiten: 245-255Band: 3Link 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 KunzeZeitschrift: Ergodic Theory and Dynamical SystemsSeiten: 1210–1229Band: 34Link 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 GerlachZeitschrift: PositivitySeiten: 875–898Band: 17Link 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 NittkaZeitschrift: Journal of Mathematical Analysis and ApplicationsSeiten: 763–774Band: 388Link 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

#### Theses

• 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)