An introduction to large deviations

WS 2019/2020


Lectures: Thursday 09:15 - 10:45, HS 19 (Augustusplatz 10)

Office hours: appointment via email



The course is oriented on master and PhD students as the first look at the theory of large deviations. It is an extension of the course taught at Jilin University (see here)



TOPICS


  • Cramer's theorem;
  • Notion of large deviation principle;
  • Contraction principle;
  • Schilder's theorem (LDP for Brownian motion);
  • Friedlin-Wentzell theory (LDP for SDE);
  • Gärtner-Ellis's theorem;
  • Varadhan's lemma;
  • Exponential tightness and weak LDP
  • Application of LDP


    • BASIC LITERATURE


  • Frank den Hollander. "Large deviations"
  • Olav Kallenberg "Foundations of modern probability" 2002, Second Edition


    • ADDITIONAL LITERATURE


  • Peter Mörters "Large deviation theory and applications"
  • Scott Robertson "Large Deviation Principles"
  • Amir Dembo and Ofer Zeitouni "Large deviations techniques and applications"
  • Firas Rassoul-Agha and Timo Seppäläinen "A course on large deviations with an introduction to Gibbs measures"


    • LECTURE NOTES


    Notes (Lectures 1-9)



    COURSE JOURNAL


  • Oct 17 - Introduction and some examples
  • Oct 24 - Cramer's theorem
  • Nov 14 - Large deviation principle for Gaussian vectors
  • Nov 21 - Lower semi-continuity and goodness of rate functions
  • Nov 28 - Lower semi-continuity and goodness of rate functions (Part II)
  • Nov 5 - Weak LDP and exponential tightness
  • Dec 12 - LDP for Brownian motion
  • Dec 19 - Cameron-Martin formula
  • Jan 9 - Proof of Schilder theorem
  • Jan 16 - Contraction principle and Freidlin-Wentzell theory
  • Jan 23 - Some applications of large deviations