General Prerequisites: Part A Probability and Part A Integration are required. B8.1 (Measure, Probability and Martingales), B8.2 (Continuous Martingales and Stochastic Calculus) and C8.1 (Stochastic Differential Equations) are desirable, but not essential.
Course Overview: The convergence theory of probability distributions on path space is an essential part of modern probability and stochastic analysis allowing the development of diffusion approximations and the study of scaling limits in many settings. The theory of large deviation is an important aspect of limit theory in probability as it enables a description of the probabilities of rare events. The emphasis of the course will be on the development of the necessary tools for proving various limit results and the analysis of large deviations which have universal value. These topics are fundamental within probability and stochastic analysis and have extensive applications in current research in the study of random systems, statistical mechanics, functional analysis, PDEs, quantum mechanics, quantitative finance and other applications.
Learning Outcomes: The students will understand the notions of convergence of probability laws, and the tools for proving associated limit theorems. They will have developed the basic techniques for the establishing large deviation principles and be able to analyze some fundamental examples.
Course Synopsis: 1) (2 lectures) We will recall metric spaces, and introduce Polish spaces, and probability measures on metric spaces. Weak convergence of probability measures and tightness, Prohorov's theorem on tightness of probability measures, Skorohod's representation theorem for weak convergence.
2) (2 lectures) The criterion of pre-compactness for distributions on continuous path spaces, martingales and compactness.
3) (4 hours) Skorohod's topology and metric on the space \(D[0,\infty)\) of right-continuous paths with left limits, basic properties such as completeness and separability, weak convergence and pre-compacness of distributions on \(D[0,\infty)\). D. Aldous' pre-compactness criterion via stopping times.
4) (4 lectures) First examples - Cramér's theorem for finite dimensional distributions, Sanov's theorem. Schilder's theorem for the large deviation principle for Brownian motion in small time, law of the iterated logarithm for Brownian motion.
5) (4 lectures) General tools in large deviations. Rate functions, good rate functions, large deviation principles, weak large deviation principles and exponential tightness. Varadhan's contraction principle, functional limit theorems.
