TY - JOUR

T1 - Continuity in κ in SLEκ theory using a constructive method and Rough Path Theory

AU - Beliaev, Dmitry

AU - Lyons, Terry J.

AU - Margarint, Vlad

N1 - Funding Information:
T. L and V. M were supported by ERC Advanced Grant (Grant Agreement No.291244 Esig), V. M. was funded by EPSRC grant 1657722, D. B. and V. M. were partially funded by EPSRC Fellowship EP/M002896/1. V. M. acknowledges also the support of NYU-ECNU Institute of Mathematical Sciences at NYU Shanghai. We thank Huy Tran and Yizheng Yuan for reading the draft and offering useful suggestions.
Publisher Copyright:
© Association des Publications de l’Institut Henri Poincaré, 2021

PY - 2021/2

Y1 - 2021/2

N2 - 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+

AB - 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+

KW - Continuity in κ

KW - Rough Path Theory

KW - Schramm–Loewner Evolutions

UR - http://www.scopus.com/inward/record.url?scp=85104244541&partnerID=8YFLogxK

U2 - 10.1214/20-AIHP1084

DO - 10.1214/20-AIHP1084

M3 - Article

AN - SCOPUS:85104244541

VL - 57

SP - 455

EP - 468

JO - Annales de l'institut Henri Poincare (B) Probability and Statistics

JF - Annales de l'institut Henri Poincare (B) Probability and Statistics

SN - 0246-0203

IS - 1

ER -