# MA4L3 Large Deviation Theory

Not Running 2019/20

Lecturer:

Term(s): Term 1

Status for Mathematics students: List C

Commitment: 30 Lectures

Assessment: 85% Exam and 15% Homework

Prerequisites:

MA359 Measure Theory (or equivalently any of ST342 Maths of Random Events or MA3H2 Markov Processes and Percolation Theory)

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:

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

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

 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