Termine Angewandte Mathematik Münster

Abstract: Let G be a measurable semigroup. Consider a stationary ergodic Markov chain ? = (?n)n ?? taking values in G. We are interested in conditions on ? ensuring that for every measure preserving and ergodic action {Tg}g G? of G on a probability space (X,?) and every f ? L1(X) the random averages (f + f T? ?0 + ... + f T? ?0...?N-2)/N converge a.s. to the integral ? f d?. Considering the shift map S on the path space (?,P) of the Markov chain we may associate to every such action {Tg}g G? the skew product T of S and {Tg}g G? . This allows us to reduce the above problem to the question whether for every ergodic action {Tg}g G? of G the respective skew product T is also ergodic. For iid processes ? this question is answered in the affirmative by a classical result of Kakutani. As a consequence on obtains Kakutani?s well known random ergodic theorem, which has found wide applications in the study of random walk dynamics. For finite state Markov chains Bufetov introduced the condition of strict irreducibility and showed that Markov chains satisfying this condition also satisfy the above property. Such Markov chains arise for instance naturally from Markov codings of certain word hyperbolic groups such as free groups and Fuchsian groups. In this talk I will explain how the notion of strict irreducibility can be extended to Markov chains with arbitrary state spaces. The main result I will present shows that the strictly irreducible Markov chains are precisely the Markov chains satisfying the above property. This provides us with random ergodic theorems for Markovian averages of general semigroup actions generalizing Kakutani?s as well as Bufetovs results. The talk is based on joint work with Pablo Lummerzheim and Felix Pogorzelski.

TBA

Dieses Praktikum richtet sich vor allem an Studierende der Mathematik, Physik, Chemie und Biologie. In interdisziplinären Kleingruppen werden anwendungsorientierte Fragestellungen bearbeitet.

Viele physikalische, chemische oder auch biologische Prozesse beinhalten mehrere zeitliche und räumliche Skalen. Die zugrunde liegenden Prozesse weisen zudem häufig eine nichtlineare Dynamik auf, die das makroskopische Verhalten maßgeblich bestimmt. Die Modellierung solcher Prozesse stellt aufgrund der komplexen Dynamik eine große Herausforderung dar, sowohl bei der physikalischen, chemischen oder biologischen Formulierung als auch der mathema- tischen/numerischen Behandlung.

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