Brownian Motion (Spring, 2008)

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

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

January 15th.

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


    Brownian Motion is THE fundamential process in probability theory.

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

    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.