This page contains lecture notes and other material for the courses I have taught.
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 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 to tackle a class
of rather simple non-linear equations.
This is an advanced course aimed mostly at postgraduate students with a good
background in probability theory and functional analysis.
It was tought in the second term of the 2007/08 academic year.
Lecture Notes (Still in progress...)
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
Exercises for week 2
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