Path properties of Brownian motion
MATH 296M: SPRING 2004

 

time: MWF 10:00 AM-10:50 AM
place: MS 6627
lecturer: Marek Biskup (MS 6617B)
office hours: WALK IN

Course outline: first announcement

The general goal of this course is to study sample path properties of one-dimensional Brownian motion. This topic is interesting both for probability theory (sample paths are random) and analysis (sample paths are very rough). The emphasis will lie in-between the two fields; we will use probability every now and then but we will also use a lot of real analysis. The two-dimensional Brownian motion will the subject of a course (MATH 285k) that I will teach in the Fall 2004.

Here is a somewhat more detailed plan (subject to changes) for this quarter; here is the schedule which details who is lecturing and when (also subject to regular updates).

Materials: (preliminary selection)

  1. Y. Peres et al, Lecture notes on Brownian motion (to be updated)
  2. I. Karatzas and S.E. Shreve, Brownian Motion and Stochastic Calculus, Graduate Texts in Mathematics, vol. 113, Springer-Verlag, New York, 1991.
  3. K. Burdzy, On Nonincrease of Brownian Motion, Ann. Probab. 18 (1990), no. 3, 978-980.
  4. B. Davis, On Brownian slow points, Z. Wahrsch. Verw. Gebiete 64 (1983) 359-367.
  5. B. Davis and E. Perkins, Brownian slow points: The critical case, Ann. Probab. 13 (1985), no. 3, 779-803.
  6. H.P. McKean, Hausdorff-Besicovitch dimension of Brownian motion paths, Duke Math J. 22 (1955) 229-234.
  7. S. Orey and S.J. Taylor, How often on a Brownian path does the law of iterated logarithm fail, Proc. London Math. Soc. (3) 28 (1974) 174-192.
  8. K. Burdzy, Cut points on Brownian paths, Ann. Probab. 17 (1989), no. 3, 1012-1036.
  9. E. Perkins, On the Hausdorff dimension of the Brownian slow points, Z. Wahrsch. Verw. Gebiete 64 (1983), no. 3, 369-399.

Prerequisites:

Real analysis in the extent of MATH 245AB will be used without apology. Complex analysis may be used later in the course; we will brush up the needed facts. Good acquaintance with basic (graduate) probability 275A will be necessary; however, if you are willing to accept some things as a fact, you can get through by simultaneously taking 275A. The knowledge of the 275C material is not required (nor of 275B).

Conditions for getting credit:

I am not permitted to give you credit for attending unless you give at least two one-hour presentations. I am sure that, once we get going, this requirement will be a piece of cake for you to fulfill.

In order to keep the course lively and understandable for everybody, there will be some emphasis on the style of your presentations. The people who have been thru the "TA-training" with me know what this amounts to; others will learn this as time goes on.