On the existence of SLE trace: finite energy drivers and non-constant $\kappa$

TL;DR

This paper investigates the existence of Schramm-Loewner Evolution (SLE) traces by analyzing finite energy drivers and rough paths, extending classical results to non-constant  and perturbed drivers using advanced stochastic calculus techniques.

Contribution

It introduces a new framework for establishing SLE trace existence with finite energy drivers and rough paths, accommodating non-constant  and drift perturbations.

Findings

Finite energy paths serve as natural drivers with simple estimates.

Representation of the inverse Loewner flow derivative via rough and Follmer integrals.

Existence of SLE trace established for drivers with finite energy and non-constant .

Abstract

Existence of Loewner trace is revisited. We identify finite energy paths (the "skeleton of Wiener measure") as natural class of regular drivers for which we find simple and natural estimates in terms of their (Cameron--Martin) norm. Secondly, now dealing with potentially rough drivers, a representation of the derivative of the (inverse of the) Loewner flow is given in terms of a rough- and then pathwise F\"ollmer integral. Assuming the driver within a class of It\^o-processes, an exponential martingale argument implies existence of trace. In contrast to classical (exact) SLE computations, our arguments are well adapted to perturbations, such as non-constant $\kappa$ (assuming $<2$ for technical reasons) and additional finite-energy drift terms.