Quasi-Sure Stochastic Analysis through Aggregation and SLE${}_{\kappa }$ Theory

Abstract

We study SLE${}_{\kappa }$ theory with elements of Quasi-Sure Stochastic Analysis through Aggregation. Specifically, we show how the latter can be used to construct the SLE${}_{\kappa }$ traces quasi-surely (i.e. simultaneously for a family of probability measures with certain properties) for $\kappa \in \mathcal{K}\cap {\mathbb{R}}_{+}\setminus ([0,\epsilon )\cup \{8\})$, for any $\epsilon >0$ with $\mathcal{K}\subset {\mathbb{R}}_{+}$ a nontrivial compact interval, i.e. for all $\kappa$ that are not in a neighborhood of zero and are different from $8$. As a by-product of the analysis, we show in this language a version of the continuity in $\kappa$ of the SLE${}_{\kappa }$ traces for all $\kappa$ in compact intervals as above.