Math 778 — Stochastic Processes

Spring 2003

Instructor: Greg Lawler

Time: TR 11:40-12:55

Room: MT 206

Topic: The Stochastic Loewner Evolution (SLE)

An introduction SLE and its applicationa to random two-dimensional processes. Topics include:

• Definition and properties of SLE (SLE is obtained by solving the Loewner equation with a Brownian motion input). The deterministic Loewner equation is being discussed this fall in Math 613.
   
• How to show the outer boundary of planar Brownian motion has Hausdorff dimension 4/3 This includes discussion of the relationship between the "intersection exponents" of Brownian motion and dimensions of exceptional sets.
   
• Why conformal invariance of percolation clusters implies that the boundaries are given by SLE paths.
   
• The "restriction property" and why this tells us what self-avoiding walks "should" look like in the limit

I will be assuming material that I do this semester in Math 613. The material from both courses will become a book; the notes for Math 613 should be available by the end of this semester, so people who wish to take Math 778 without having Math 613 can manage with extra reading.