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 Masterslevel 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 Φ^{4}_{2} Euclidean quantum field theory. It also includes short proof of Nualart and Peccati's fourth moment theorem.
PDF file of the Lecture Notes
Introduction 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 Notes
A Course on Rough Paths
This book is somewhat complementary to existing works on the theory of rough paths.
PDF file of the book
Introduction 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 infinitedimensional 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 nonlinear 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 selfcontained 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 Schrödingertype 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 longtime 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 PerronFrobenius 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
