Schramm-Loewner Evolution

TL;DR

These notes provide an updated, perspective-shifting exposition on Schramm-Loewner Evolution (SLE), emphasizing its measure-theoretic formulation and connections to lattice models, without covering convergence from discrete processes.

Contribution

The notes introduce SLE as a finite measure on paths and utilize Girsanov's theorem more extensively, offering a new perspective distinct from previous expository works.

Findings

SLE defined as a finite measure on paths, not necessarily probability.

Use of Girsanov's theorem to study martingales related to SLE.

Focus on continuous SLE without convergence proofs from discrete models.

Abstract

This is the first expository set of notes on SLE I have written since publishing a book two years ago [45]. That book covers material from a year-long class, so I cannot cover everything there. However, these notes are not just a subset of those notes, because there is a slight change of perspective. The main differences are: o I have defined SLE as a finite measure on paths that is not necessarily a probability measure. This seems more natural from the perspective of limits of lattice systems and seems to be more useful when extending SLE to non-simply connected domains. (However, I do not discuss non-simply connected domains in these notes.) o I have made more use of the Girsanov theorem in studying corresponding martingales and local martingales. As in [45], I will focus these notes on the continuous process SLE and will not prove any results about convergence of discrete processes to SLE. However, my first lecture will be about discrete processes -- it is very hard to appreciate SLE if one does not understand what it is trying to model.