Functional Analysis and Stochastic Analysis Group


General information

Listed below are the courses and lectures provided by our group during summer term 2020.

If you are interested to attend a reading course this term, or if you wish to write a thesis within our group, please feel free to contact your favoured supervisor in person or by e-mail.

Important links

  • KIS: dates of lectures and tutorials
  • URM: registration for tutorials
  • OpenOLAT: course materials and further information (access codes are made available in the first lecture)

Maß- und Integrationstheorie

Inhalte

  • Mengensysteme/-ringe (σ-Algebren)
  • Maße, Lebesgue-Maß
  • Satz von Carathéodory, Satz von Radon-Nikodým
  • messbare Funktionen, Approximationssatz
  • Lebesgue-Integral, Lp-Räume, Konvergenzsätze, Transformationssatz
  • Produktmaße, Satz von Fubini.

Kontaktzeit

2 SWS / 30h Vorlesung
1 SWS / 15h Übung

Inhaltliche Voraussetzungen

Lehrveranstaltungen „Grundlagen der Mathematik I+ II“

Functional Analysis

Content

  • Hahn-Banach theorem and its applications
  • Baire category theorem and its applications (uniform boundedness principle, Banach-Steinhaus theorem, open mapping theorem, inverse mapping theorem, closed graph theorem)
  • weak convergence (Banach-Alaoglu theorem, reflexive Banach spaces, lemma of Mazur and its applications)
  • projections (closed complement theorem)
  • bounded operators (adjoint operators, spectrum, resolvent, normal operators)
  • compact operators (Fredholm operators, Fredholm alternative and its applications, spectral theorem (Riesz-Schauder) and applications to normal operators)

contact time

4 SWS / 60 h lecture
2 SWS / 30 h exercise classes

substantive prerequisites

content of the introductory lecture "Einführung in die Funktionalanalysis" as well as concepts from "Maß- und Integrationstheorie"

Operator Semigroups and Applications to PDE

Content

  • Definitionen, Generatoren, Resolventen, Beispiele, 
  • Hille-Yosida Theorem, Lumer-Phillips Theorem, 
  • Kontraktions-Halbgruppen, Analytische Halbgruppen, Operator-Gruppen, 
  • Approximationen, Störungen, 
  • Anwendungen auf Partielle Differentialgleichungen (u.a. Wärmeleitungsgleichungen, Wellengleichungen, Schrödinger-Gleichungen).

contact time

4 SWS / 60 h lecture
2 SWS / 30 h exercise classes

substantive prerequisites

content of the lectures "Functional Analysis"

Seminars in summer

Our group offers the following seminars during summer term 2020:

Seminar Analysis and Partial Differential Equations

Lectures in winter 2019/20

Our group offered the following lectures during winter term 2019/20:

Einführung in die Funktionalanalysis

Inhalte

  • Beispiele für Banachräume und Hilberträume
  • Kompaktheit, Heine-Borel, Arzelà-Ascoli
  • beschränkte lineare Operatoren, adjungierte Operatoren, Neuman-Reihe
  • Orthogonalität, Hilbertraum-Basis, Riesz-Darstellung, Lax-Milgram, selbstadjungierte Operatoren, Spektraltheorie

Kontaktzeit

2 SWS / 30 h Vorlesung
1 SWS / 15 h Übung

Inhaltliche Voraussetzungen

Lehrveranstaltungen "Grundlagen der Mathematik I + II"

Angebotsturnus

Die Vorlesung wird jedes Jahr im Wintersemester angeboten.

Hier geht es zum KIS-Eintrag:
Einführung in die Funktionalanalysis (Vorlesung)
Einführung in die Funktionalanalysis (Übung)

Hier geht es zum OLAT-Kurs:
TUK Einführung in die Funktionalanalysis WS 18/19

Probability Theory

Content

  • notions of convergence (in probability, almost surely, weak convergence, Lp-convergence, convergence in distribution)
  • characteristic functions
  • sums of independent random variables
  • strong law of large numbers, variants of the central limit theorem
  • conditional expectation
  • discrete time martingales
  • Brownian motion

contact time

4 SWS / 60 h lecture
2 SWS / 30 h exercise classes

substantive prerequisites

content of the lectures "Stochastische Methoden" and "Maß- und Integrationstheorie"

External resources

White Noise Analysis

Content

  • Introduction to the basics of distribution theory with specific focus on tempered distributions
  • Construction of the White Noise space (Minlos theorem, chaos-decomposition, T-transform, S-transform, Ito-Wiener-Segal isomorphism)
  • Introduction of test function spaces and spaces of generalised functions of White Noise Analysis (Hida and Kondratiev spaces)
  • Applications to Feyman path integrals and stochastic PDE

contact time

4 SWS / 60 h lecture
2 SWS / 30 h exercise classes

substantive prerequisites

Content of the introductory lecture "Einführung in die Funktionalanalysis" and "Maß- und Integrationstheorie"

Seminars in winter

Our group offers the following seminars during winter term 2018/19:

Potential Theory and Stochastic Analysis via Dirichlet forms

The seminar will take place on Tuesdays from 13:45 to 15:15 in room 48-519

First meeting: October 23, 2018

Content

In this seminar it is planned to start with developing some analytic potential theory of Dirichlet forms. We will consider so-called excessive functions and introduce an "intrinsic" notion of exceptional sets corresponding to Dirichlet forms. Having these tools in hand, we will focus on quasi-continuity of functions. Then we revisit the theory of Dirichlet forms from a probabilistic point of view. The goal is to explain how Dirichlet forms are associated properly with Markov processes. In order to do so, the notion of a quasi-regular Dirichlet form plays a crucial role. Providing a class of examples for the analytically and probabilistically studied objects will round off the seminar. It is planned to proceed along the contents of the first book from the list of references. The strength of the theory of Dirichlet forms is given by the fact that this mathematical tool is situated in a vast interdisciplinary area which includes analysis and probability theory. Therefore, applications can be found in research areas like Partial Differential Equations, Mathematical Physics (Quantum (Field) Theory, Statistical Physics), Stochastic (Partial) Differential Equations and Stochastic Analysis. Historically, its roots are in the interplay between ideas of analysis (calculus of variations, boundary value problems, potential theory) and probability theory (Brownian motion, stochastic processes, martingale theory).

Literature

  • Z.-M. Ma und M. Röckner, Introduction to the Theory of (Non-Symmetric) Dirichlet Forms, Springer, Berlin, 1992
  • M. Fukushima, Dirichlet Forms and Markov Processes, North-Holland, Amsterdam, Oxford, New York, 1980; 
  • M. Reed und B. Simon, Functional Analysis I and II, Academic Press, 1975.

Prerequisite

Lecture 'Functional Analysis'

Performance record

Certificate for presentation of a talk

Zum Seitenanfang