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Mathematics Institute

MA4L3 Content

Content:

  • Basic understanding of large deviation techniques (definition, basic properties, Cramer’s theorem, Varadhan’s lemma, Sanov’s theorem, the Gärtner-Ellis Theorem).
  • Large deviation approach to Gibbs measure theory (free energy; entropy; variational analysis; empirical process; mathematics of phase transition).
  • Large deviation theory for stochastic processes and its connections with PDEs (Fleming semi group; viscosity solutions; control theory).
  • Applications of large deviation theory (at least one of the following list of topics: interface models; pinning/wetting models; dynamical systems; decay of connectivity in percolation; Gaussian Free Field; Free energy calculations; Wasserstein gradient flow; renormalisation theory (multi-scale analysis)).

Aims:

  • Basic understanding of large deviation theory (rate function; free energy; entropy; Legendre-transform).
  • Understanding that large deviation principles provide a bridge between probability and analysis (PDEs, convex and variational analysis).
  • Large deviation theory as the mathematical foundation of mathematical statistical mechanics (Gibbs measures; free energy calculations; entropy-energy competition).
  • Understanding large deviation in terms of the nonlinear Fleming semi group and its links to control theory.
  • Discussion of the role of large deviation methods and results in joining different scales, e.g. as the micro-macro passage in interacting systems.
  • Connection of large deviation theory with stochastic limit theorems (law of large numbers; ergodic theorems (time and space translations); scaling limits).

Objectives: By the end of the module students should be able to:

  • Derive basic large deviation principles
  • Be familiar with the variational principle and the large deviation approach to Gibbs measure
  • Distinguish all three level of large deviation
  • To calculate Legendre-Fenchel transform for most relevant distributions
  • Understand basic variational problems
  • Be familiar with some application of large deviation theory
  • Link basic large deviation principle for stochastic processes to PDEs
  • Compute of rare probabilities via large deviation rate functions given as variational problems in analysis and PDE theory. Be able to use Legendre-transform techniques, basic convex analysis and Laplace integral methods.

  • Understand the role of free energy calculations and representations in analysis (PDEs and control problems and variational problems). Be able to provide a variational description of Gibbs measures.

  • Be able to analyse the minimiser of large deviation rate functions of basic examples and to provide interpretation of the possible occurrence of multiple minimiser.

  • Explain the role of the free energy in interacting systems and its link to stochastic modelling. Be able to provide different representations of the free energy for some basic examples.

  • Be able to estimate probabilities for interacting systems using Laplace integral techniques and basic understanding of Gibbs distributions.

  • Apply large deviation theory to one topic from the following list: interface models; pinning/wetting models (random walk models); dynamical systems; decay of connectivity in percolation; Gaussian Free Field; Free energy calculations; Wasserstein gradient flow; renormalisation theory (multi-scale analysis).

Books: We won’t follow a particular book and will provide lecture notes. The course is based on the following three books:

[1] Frank den Hollander, Large Deviations (Fields Institute Monographs), (paperback), American Mathematical Society (2008).

[2] Amir Dembo & Ofer Zeitouni, Large Deviations Techniques and Applications (Stochastic Modelling and Applied Probability), (paperback), Springer (2009).

[3] Jin Feng and Thomas G. Kurtz, Large Deviations for Stochastic Processes, American Mathematical Society (2006).

Other relevant books and lecture notes:

[a] Hans-Otto Georgii, Gibbs Measures and Phase Transitions, De Gruyter (1988).

[b] Stefan Adams, Lectures on mathematical statistical mechanics, Communications of the Dublin Institute for Advanced Studies Series A (Theoretical Physics), No. 30 , available online http://www2.warwick.ac.uk/fac/sci/maths/people/staff/stefan adams/lecturenotestvi/cdias-adams-30.pdf

[c] Stefan Adams, Large Deviations for Stochastic Processes, EURANDOM reports 2012-25, (2012); available online http://www.eurandom.tue.nl/reports/2012/025-report.pdf