Brownian Motion (Spring, 2008)

There is an email list for the course. If you want to be on it, send me an email!!!!!!!!!!!!

READING PERIODS:
This course will go the third and fourth reading periods.
(It is possible we don't go through the entire fourth reading period.)

TIMES AND PLACE (for the third reading period):
Tuesdays and Thursdays 13:15-15:00, Room: MV:L14

FIRST MEETING TIME:
January 15th.

POINTS:
7 1/2 "new style" points (5 "old style" points)

WHAT IS BROWNIAN MOTION?

Brownian Motion is THE fundamential process in probability theory.

It is a Markov (even Levy) Process, a martingale, and a Gaussian process.
It is "the" example of a continuous time, continuous space random walk.
It is universal (in a sense that the normal distribution is universal due to the
Central Limit Theorem)
It is intimately connected to potential theory (the solution of the Dirichlet
problem can be described in terms of Brownian motion )
One can perform exact calculations for Brownian motion
Due to recent important developments (SLE), it is known that Brownian Motion
is what is lying behind various exact critical exponents in statistical mechanics.

WHAT WILL WE TRY TO COVER?

In this course, we will cover various aspects of Brownian motion. The planned topics
are below (although this plan is potentially overly ambitious).

The existence of Brownian Motion
Some of its sample path properties such as the fact that it is almost surely nowhere
differentiable (and a discussion that its "Holder exponent" is 1/2), as well as many other properties.
The structure of the zero set of Brownian motion (it is a perfect set of dimension 1/2)
The law of the Iterated Logarithm
Levy's theorem on the exact modulus of continuity of Brownian motion; this leads easily,
via a Baire Category argument, to the existence of "fast points" (which yields a multifractal
spectrum which will only be mentioned).
Arc sine laws
Exact passage time distribution
Skorohod embedding
Donsker's Invariance Principle (a.k.a., the functional central limit theorem)
The solution of the classical Dirichlet problem in terms of Brownian motion
The fact that BM has double points in 3 dimensions and not in 5 dimensions.
(It will be also outlined why the critical dimension for self-intersection is 4.)
Description of the first passage times process in terms of Poisson random
measures (this is due to the fact that this process is a Levy process which are
all described in terms of so-called Levy measures). (If time permits, this will all
be related to the notion of "local time" for BM which also connects to Ito's
excursion representation of BM).

INTENDED AUDIENCE: This course is intended for graduate students in mathematics
and mathematical statistics. (It could also be taken by advanced undergraduate students
with very good mathematical backgrounds). Faculty are also of course welcome.

PREREQUISITES: While it is always hard to say exactly what is needed, it is expected that
you have taken (or at least know) either "Sannolikheter och väntevärden" or "Integrationsteori".
In addition, you should have had reasonable exposure to probability theory, ideally at the
graduate level (like in the first course above). (This could be replaced by a strong background
in analysis.) Talk to me if you are unsure.

COURSE LITERATURE : There will be no official book for the course but "very much" of what
we do can be found here which is a draft of the book by Morters and Peres on BM. here

Other good sources (varying in difficulty) are

Durrett: Chapter 7 in Probability: Theory and Examples
Karatzas, I. and Shreve, S. : Brownian Motion and Stochastic Calculus, Springer, 1994.
Revuz, D. and Yor, M. : Continuous Martingales and Brownian Motion, Springer
Rogers, L.C.G. and Williams, D. : Diffusions, Markov Processes and Martingales, Vol. I.

EXAMINATION: There will be home-assignments, and an oral examination at the end.
Graduate courses are graded on a pass-fail basis. If you are not a graduate student and
require a grade other than pass-fail, this must be discussed with me.

EXAMINATOR: Jeff Steif
email: steif"at"math.chalmers.se

Homeworks:
Homework 1 Due February 6th
Homework 2 Due February 28th
Homework 3 Due April 10

Below is a BRIEF STETCH!! of SOME of what we did and SOME of what we will do. It is NOT meant at all to be exhausive!! The official material for the course is that covered in the lectures and so if you miss classes, it is best to get notes from others. All references below will refer to here.

Here is an outline of SOME things that we have done :
Overview of Brownian motion.
Theorem 1.25 although our proof is a different.
Theorem 1.27.
limsup_{t\to\0} B_t/t^{1/2} =\infty a.s.
A special case of Theorem 2.7.
Theorem 1.8. and Lemma 1.6 (which are scale invariance and time inversion of BM)
Prop 1.23 (with a different proof).
We have shown that there is a process having the same finite dimensional distributions as BM but not dealt with the continuity of the paths.
Completing the construction of BM. We will not follow the construction in MP but follow another construction (which is longer but has many educational advantages) which can be found in
(1) Durrett: Chapter 7 in Probability: Theory and Examples or (2). Karatzas, I. and Shreve, S. : Brownian Motion and Stochastic Calculus, Springer, 1994. (There will be a handout for this construction since it is not in MP.)
We showed that BM is a measurable process.
Stopping times, the markov and strong markov property (pages 43-49).
Elementary properties of the zero set of BM (E.g., it is a perfect set, meaning there are no isolated points, and hence uncountable.) (page 53).
Hausdorff and Minkowski dimension (97-103 or so but not everything and a weaker version of Theorem 4.30 on page 110)
Hausdorff dimension of the path (range) of a d-dimensional path is 2 for all d>= 2. This is a special case of Proposition 4.38 on page 116.
Hausdorff dimension of the zero set is 1/2. (Page 118-119). This will use an important result of Paul Levy, Theorem 2.29.
Law of the Iterated Logarithm (Page 125).
Skorokhod embedding theorem (Theorem 5.12). But we will follow the development in Durrett's book
(pages 402-405) very closely where we also get as corollary a proof of the Central Limit theorem (CLT)
without using characteristic functions.
Pages 138-139 of MP gives an application of the embedding theorem to Donsker's invariance principle, which is a far-reaching generalization of the CLT. We will formulate and state this result without proving it. We will then give some applications of it to random walk.
One of the applications is one of the arc-sign laws which we will first prove for BM before applying it. For the arc-sign law for BM, we will follow pages 395-396 in Durrett's book (an alternative proof is on page 142 in MP). The application of Donsker's theorem to random walk is then given in Prop 5.21 of MP.
The Dirichlet problem and its solution via BM. More or less pages 69-73. Then applications to recurrence/transience properties of BM, more or less pages 75-76.

We have finished the Dirichlet problem stuff other than the key fact that our proposed solution
in terms of BM is continuous to the boundary at 'nice points'. We show the exterior cone
condition is sufficient. This is in MP middle of page 72 to 73.

We moved next into some self intersection properties of BM. Here I will not be following
the book presentation so closely but will do instead the classical proofs.

We first proved from 'first principles' that in 5 or more dimensions, BM does not hit itself.
We follow the original proof in
Kakutani, Shizuo On Brownian motions in $n$-space.
Proc. Imp. Acad. Tokyo 20, (1944). 648--652.
(I mailed out the PDF file for this).
Next, we did some general facts about the second moment method.
Here is a 4 page summary.

Here is an outline of SOME things that we will do next : All we have left is aversion of Kakutani's theorem which tells us which sets 'are hit by a BM'
with positive probability. Although we will do things less quantitative, we will more or
less prove the if (and the only if if we have time) of Theorem 8.19 on page 228 for d at least 3.
But the proof might be different.

From this last result and the fact that BM paths have dimension two and the other results we derived about Hausdorff dimension, we easily get that BM hits itself in 3 d.