Inhalt der Veranstaltung:
One of the key questions in the theory of Markov processes (e.g. diffusion or Markov chains) is behaviour of a process on long time scales, especially the convergence towards equilibrium. A framework to study this question is the theory of Markov semigroups that is related to the study of second order differential operators and certain inequalities related to these differential operators.
The seminar will give a high-level overview of Markov semigroups and related functional inequalities, such as spectral gap inequalities and (logarithmic) Sobolev inequalities, and try to shed some light on the question how functional inequalities can be used to analyse and understand the qualitative properties of a Markov process.

URL für die Veranstaltungswebsite: http://numerik.mi.fu-berlin.de/wiki/WS_2015/Seminar_Semigroups.php

Zielgruppe: Advanced students with interests in stochastics and functional analysis.

Voraussetzungen: Analysis I-III, Stochastik I-II

Literatur:

1. D. Bakry. On Sobolev and logarithmic Sobolev inequalities for Markov semi-groups. In New trends in stochastic analysis (Charingworth, 1994), pages 43-75, River Edge, NJ, 1997. Taniguchi Symposium, World Sci. Publishing.
2. E. B. Davies. Heat kernels and spectral theory. Cambridge University Press, Cambridge, 1990.
3. A. Guionnet and B. Zegarlinski. Lectures on Logarithmic Sobolev Inequalities. In: Séminaire de Probabilités, XXXVI, Lecture Notes in Mathematics Volume 1801, pp 1-134, 2003.
4. L. Gross. Logarithmic Sobolev inequalities. Amer. J. Math., 97, pp. 1061-1083 , 1975.
5. A. Jüngel and D. Matthes, Entropiemethoden für nichtlineare partielle Differentialgleichungen, Internationale Mathematische Nachrichten, vol. 209, pp. 1-14, 2008.