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


Beliaev, D
Lyons, T
Margarint, V


L'Institut Henri Poincare, Annales B: Probabilites et Statistiques

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Questions regarding the continuity in $\kappa$ of the $SLE_{\kappa}$ 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
$\kappa$ we use the same Brownian motion. It is very natural to assume that
with probability one, $SLE_\kappa$ depends continuously on $\kappa$. It is
rather easy to show that $SLE$ is continuous in the Carath\'eodory 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 $\kappa_j\to\kappa \in (0,
8/3)$, for almost every Brownian motion $SLE_\kappa$ 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
$SLE_{\kappa}$ traces for varying parameter $\kappa \in (0, 8/3)$. 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 $\sqrt{\kappa}B_t$ when started away from the
origin are continuous in the $p$-variation topology in the parameter $\kappa$,
for all $\kappa \in \mathbb{R}_+$

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Journal Article