Continuity in κ in SLEκ theory using a constructive method and Rough Path Theory

Dmitry Beliaev, Terry J. Lyons, Vlad Margarint

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1 Citation (Scopus)

Abstract

Questions regarding the continuity in κ of the SLEκ traces and maps appear very naturally in the study of SLE. In order to study the first question, we consider a natural coupling of SLE traces: for different values of κ we use the same Brownian motion. It is very natural to assume that with probability one, SLEκ depends continuously on κ. It is rather easy to show that SLE is continuous in the Carathéodory sense, but showing that SLE traces are continuous in the uniform sense is much harder. In this note we show that for a given sequence κj → κ ∈ (0,8/3), for almost every Brownian motion SLEκ traces converge locally uniformly. This result was also recently obtained by Friz, Tran and Yuan using different methods. In our analysis, we provide a constructive way to study the SLEκ traces for varying parameter κ ∈ (0,8/3). The argument is based on a new dynamical view on the approximation of SLE curves by curves driven by a piecewise square root approximation of the Brownian motion. The second question can be answered naturally in the framework of Rough Path Theory. Using this theory, we prove that the solutions of the backward Loewner Differential Equation driven by √κBt when started away from the origin are continuous in the p-variation topology in the parameter κ, for all κ ∈ R+

Original languageEnglish
Pages (from-to)455-468
Number of pages14
JournalAnnales de l'institut Henri Poincare (B) Probability and Statistics
Volume57
Issue number1
DOIs
Publication statusPublished - Feb 2021
Externally publishedYes

Keywords

  • Continuity in κ
  • Rough Path Theory
  • Schramm–Loewner Evolutions

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