LMS-EPSRC Short
Course
Stochastic Partial Differential
Equations
Imperial College London
7-11 July 2008
Organiser: Dan Crisan
1. Course outline and prerequisites
The aim of the course is to
provide a good starting point for future researchers in SPDEs. It will assume
that the students are familiar with basic functional analysis and probability
theory, Ito calculus and PDE theory and build on this knowledge so that, by the
end of the course, the students will have an overall view of the main results,
themes and techniques of the area.
The course will consists of
three mini-lectures of five hours each on:
- Wiener Chaos Approach to SPDEs
- Applications of Malliavin Calculus
- Long time behaviour of SPDEs
and two guest lectures
given by Terry Lyons (
The following topics are
prerequisites for the course: Ito calculus, basic functional analysis and PDE
theory.
2. Details of the lectures
Wiener Chaos Approach to SPDEs
Lecturer: Boris Rozovsky (
Abstract: The course will start with an
elementary treatment Malliavin calculus and the Skorokhod Integral. It follow
on to introduce the basics of linear
SPDEs, a few examples of nonlinear SPDEs, asymptotic, and possibly
numerical aspects. The emphasis will be
on analytical issues.
Prerequisites: Ito calculus
1. SPDEs: What it is all about?
2. Explicitly Solvable SPDEs
3. Parabolic SPDEs in Sobolev Spaces
4. Wiener Chaos
5. Bi-linear Elliptic SPDEs
6. Feinman-Kac and Wiener Chaos Representations of
SPDEs.
Applications of Malliavin Calculus
to SPDEs
Lecturer: Marta Sanz-Solé (Barcelona, Spain)
Abstract: The course will start with a brief
introduction to Malliavin calculus: integration by parts and criteria for
existence and smoothness of probability laws (this part will be coordinated
with the first course). The course will concentrate on existence and smoothness
of the law for some classes of SPDEs, including the stochastic heat and wave
equations. It will conclude with some results on small time behavior of the
laws.
Prerequisites: Ito calculus
The following are the titles of the
individual lectures:
1. Integration by parts and absolute continuity of
probability laws.
2. Stochastic calculus of variations: the derivative,
divergence and Ornstein-Uhlenbeck operators on an abstract Wiener space.
3. Criteria for existence and regularity of densities.
4. Watanabe-Sobolev differentiability of SPDEs.
5. Analysis of the covariance matrix.
6. Small perturbations of the density.
Here are
the lecture notes 
of the course (revised version) and the tutorial material: Tutorial
1, Tutorial 2, Tutorial
3.
References:
- D.R. Bell: The Malliavin Calculus. Dover Publications, Inc.
- P. Malliavin: Stochastic Analysis. Grundlehren
der mathematischen Wissenschaften, 313. Springer Verlag, 1997
- D. Nualart: The Malliavin Calculus and
Related Topics. 2nd Edition. Springer Verlag, 2006
- D. Nualart: Analysis on the Wiener space
and anticipating calculus, In: Ecole d'Eté de Probabilités
de Saint Flour XXV, Lecture Notes in Math. 1690, Springer Verlag,
1998.
- M. Sanz-Solé: Malliavin Calculus with Applications to
Stochastic Partial Differential Equations, CRC Press,
- D. W. Stroock: Some Application of Stochastic Calculus
to Partial Differential Equations. In: Ecole d'Eté de
Probabilités de Saint-Flour XI-1981. P.L. Hennequin (Ed.). Lecture
Notes in Math. 976, pp. 268-380. Springer Verlag,1983.
- S. Watanabe: Lectures on Stochastic differential
equations and Malliavin Calculus. Tata Institute of Fundamental
Research.
Long-time behaviour of SPDEs
Lecturer: Martin Hairer (
Abstract: The course will review some basic
results of ergodic theory for Markov processes (Perron-Frobenius, Birkhoff's
ergodic theorem, Doob-Khasminskii, etc) with an emphasis on the interplay
between measure-theoretical and topological concepts. The regularising
properties of semigroups associated to semilinear parabolic PDEs will be
discussed. Also included will be simple proofs of the ergodicity results a la
Kuksin, Shirikian, E, Mattingly, Sinai, etc.
Prerequisites: Basic functional
analysis / PDE theory (see, for example, the following link.)
The following are the titles of the
individual lectures:
1. Basics of ergodic theory for Markov processes
2. Some existence / uniqueness / convergence
criteria
3. Hypoellipticity and long-time behaviour of
finite-dimensional diffusions
4. Infinite dimensions: what can go wrong?
5. The strong Feller property for linear and
non-linear SPDEs
6. The asymptotic strong Feller property
Cubature
Lecturer: Terry Lyons (
Abstract:
Identifying the numerical solution to a linear 2nd
order differential equation of parabolic type is equivalent to solving a
stochastic differential equation in the weak sense. In other words, it involves an integration of
a highly non linear function over path space. Work of Kusuoka, Victoir,
Litterer and Lyons has resulted in the development of interesting and
potentially powerful high-order numerical methods. The methods seem to share the benefits of
Monte-Carlo (feasible in high dimensions) and more classical numerical methods
(grids and finite elements - accurate when you can do them), and would seem to
have potential for providing accurate approximate solutions to non-linear
stochastic PDEs such as those that appear in the filtering problem.
Stochastic PDEs as regularizations
of PDEs by random noises
Lecturer: Istvan Gyongy (
Abstract:
Deterministic partial differential equations may be
ill-behaved in the sense that they have no solution, or non-unique solutions,
or solutions which do not depend continuously on initial conditions. Sometimes,
as with the famous Navier--Stokes equation in dimension 3, it is unknown
whether the equation is well-posed. Surprisingly, by introducing a small random
perturbation one can often transform these equations into well-posed stochastic
partial differential equations. Our aim is to present some prototypes of this
phenomenon.
3. Timetable
|
Time |
Monday |
Tuesday |
Wednesday |
Thursday |
Friday |
|
9.00-10.00
10.00-11.00 11.00-11.30 11.30-12.30
12.30-13.30 13.30-14.30
14.30-15.30
15.30-16.00 16.00-17.00 17.30-18.00 |
C1 C1 Coffee
break IL1 Lunch C2 C2 Tea
break C3 C3 |
C2 C2 Coffee
break T3 Lunch C1 C1 Tea
break T1 T2 |
C3 C3 Coffee
break C1 Lunch Trip |
C2 C2 Coffee
break IL2 Lunch C1 T2 Tea
break T3 T1 |
C3 C3 Coffee
break T1 Lunch T2 T3 |
Course key:
C1/T1 Wiener Chaos Approach to
SPDEs (lecture/tutorial)
C2/T2 Applications of Malliavin Calculus (lecture/tutorial)
C3/T3 Long time behaviour of SPDEs (lecture/tutorial)
IL1 Invited Lecture: Cubature
IL2 Invited Lecture: SPDEs as regularizations of PDEs by random noises
5. Venue details and
other arrangements
The
course will be held in Room 539 in the Blackett Laboratory,
