An introduction to large deviations

Mini-course at Jilin University, September 2019


Schedule of the lectures can be found here

Office hours: If you have any questions please come to my office - 411 at any time convenient for you. Usually, I am in the office from 8 a.m. to 5 p.m. If you want to be sure that you find me there please write me or Jia-nan Zhu in order to make an appointment.



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


    • 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


  • Sep 9 - Introduction and some examples
  • Sep 10 - Cramer's theorem
  • Sep 11 - Large deviation principle for Gaussian vectors
  • Sep 16 - Lower semi-continuity and goodness of rate functions
  • Sep 18 - Weak large deviation principle and exponential tightness
  • Sep 20 - Large deviation principle for Brownian motion
  • Sep 24 - Proof of Schilder's theorem
  • Sep 25 - Contraction principle and Freidlin-Wentzell theory
  • Sep 27 - Some applications of large deviations