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Lectures This page contains lecture notes and other material for some of the advanced courses I have taught.Advanced Stochastic Analysis These are lecture notes from a Masters-level course taught at the University of Warwick in spring 2016 and at Imperial College in spring 2021. It gives a gentle introduction to Malliavin calculus with two aims in mind: the probabilistic proof of Hörmander's theorem and Nelson's construction of the Φ42 Euclidean quantum field theory. It also includes short proof of Nualart and Peccati's fourth moment theorem. PDF file of the Lecture NotesIntroduction to Regularity Structures These are notes from a series of lectures given at the University of Rennes in June 2013, at the University of Bonn in July 2013, at the XVIIth Brazilian School of Probability in Mambucaba in August 2013, and at ETH Zurich in September 2013. They give a concise overview of the theory of regularity structures as exposed in this article. In order to allow to focus on the conceptual aspects of the theory, many proofs are omitted and statements are simplified. PDF file of the Lecture NotesA Course on Rough Paths This book is somewhat complementary to existing works on the theory of rough paths. PDF file of the bookIntroduction to Stochastic PDEs This course gives a survey of techniques and results in the field of stochastic partial differential equations. It starts by recalling the basics of the theory of Gaussian measures on infinite-dimensional spaces and of semigroup theory. This then allows us to proceed to the study of linear stochastic partial differential equations (stochastic heat equation, stochastic wave equation). We then build on this foundation to tackle a class of non-linear equations.It is an advanced course aimed at postgraduate students and researchers with a good background in probability theory and functional analysis. I taught it at the University of Warwick in the second term of the 2007/08 academic year and at the Courant Institute in the second term of 2008/09. It is still being polished, so please check back for updates! In the meantime, if you happen to find any typo, mathematical or factual mistake, omission, etc, please by all means let me know! PDF file of the Lecture Notes P@W course on the convergence of Markov processes These are lecture notes for a five times 1 and 1/2 hours minicourse on the convergence of Markov processes given at the University of Warwick in July 2010. Highlights that are not so easy to find in the literature are upper and lower bounds on the convergence rate to equilibrium for Markov processes that exhibit subgeometric convergence, as well as an elementary sketch of the probabilistic proof of Hörmander's theorem.PDF file of the Lecture Notes See also this link for a self-contained proof of Hörmander's theorem based on these notes. LMS course on Hypoelliptic Schrödinger type operators These are lecture notes for a six hours minicourse on the spectral properties of hypoelliptic Schrödinger-type operators (such as those arising from the Langevin equation after) given at Imperial College, London, in July 2007. These notes were typed by Piotr Ługiewicz.PDF file of the Lecture Notes LMS course on Stochastic PDEs These are lecture notes for a six hours minicourse on the ergodic theory of stochastic PDEs given at Imperial College, London, in July 2008. Parts of these notes are quite rough around the edges and give sketches of proofs and main ideas, rather than a sequence of completely rigorous steps.PDF file of the Lecture Notes Ergodic properties of Markov processes Markov processes are used to model a wide range of situations, ranging from the shuffling of a deck of cards to weather forecast predictions. In this course, we will study the long-time behaviour of such processes. Intuitively this corresponds to the following type of question: "If I prepare a system in a specific initial state and then let it evolve, how long do I have to wait until all information about this initial state is lost, and does this happen at all?" or "If I record an observable of my system over a very long period of time and compute its average, does this converge and how can I compute the limit?"The mathematical theorems that provide answers to these questions are the Perron-Frobenius theorem and the (generalised) law of large numbers. We will prove these and other related results in a fairly general setting. We will also study several techniques that allow to get more constructive results than what the general theorems provide. This is a fourth year course that was tought in the second term of the 2004/05 and the 2005/06 academic years at the University of Warwick. Lecture notes Exercises for week 2 Exercises for week 3 Exercises for week 4 Exercises for week 5 Exercises for week 6 Exercises for week 7 Exercises for week 8 Exercises for week 9 Exercises for week 10 |