Vorlesung + Übung: Stochastische Prozesse (Stochastik IV) - Details
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General information

Course name Vorlesung + Übung: Stochastische Prozesse (Stochastik IV)
Subtitle Stochastic Processes
Course number MTH-1670
Semester SS 2023
Current number of participants 24
Home institute Stochastik und ihre Anwendungen
participating institutes Institut für Mathematik
Courses type Vorlesung + Übung in category Teaching
Next date Tue., 06.06.2023 14:00 - 15:30, Room: (L-1007)
Pre-requisites Profound knowledge of measure-theoretic probability theory (e.g. bachelor modules "Stochastik I/II"). Basics might be recalled from Durrett's book (reference below).
Online/Digitale Veranstaltung Veranstaltung wird in Präsenz abgehalten.
Hauptunterrichtssprache englisch
Literaturhinweise • T.M. Liggett, Continuous Time Markov Processes, AMS 2010.
– our main textbook for this course
• L.B. Koralov and Ya. Sinai, Theory of Probability and Random Processes, Springer 2010.
– an alternative presentation of the material
• A. Klenke, Probability Theory, Springer 2014.
– an even different presentation of the material, also available in German
• P. Mörters, Y. Peres, Brownian Motion, Cambridge University Press 2010.
– specifically for the chapter on Brownian motion
• R. Durrett, Probability. Theory and Examples, 5th ed., Cambridge University Press 2019.
– contains all the basics in probability theory
Furthermore, we provide lecture notes for the chapter on Markov Processes (see under Files).
Miscellanea Overview:
I. Basic notions
II. Brownian motion
III. Markov chains
IV. Feller Processes
V. Interacting particle systems
ECTS points 9

Course location / Course dates

(L-1007) Tuesday: 14:00 - 15:30, weekly (13x)
(L-1008) Wednesday: 14:00 - 15:30, weekly (12x)
(Raum 1008/L) Thursday: 14:00 - 15:30, weekly (12x)

Module assignments


Welcome to the course Stochastic Processes!
This is a core module in probability theory.

The course presents some elements of the mathematical foundation for continuous-time stochastic processes. Central topic in this course is the theory of Feller Processes. A major theme is the Hille-Yoshida theory, which provides a correspondence between Feller Processes, semigroups and a class of linear operators (the "generators" of the processes). We will prove this correspondence, and subsequently show how properties of the operators translate into properties of the processes and vice versa. We then focus on specific examples of Feller processes that are of great importance in probability theory: Brownian motion, Lévy processes, continuous-time Markov chains and interacting particle systems.