论文标题
使用建设性方法和粗糙路径理论中的$κ$ $κ$中的$κ$中的$κ$
Continuity in $κ$ in $SLE_κ$ theory using a constructive method and Rough Path Theory
论文作者
论文摘要
关于$SLE_κ$痕迹和地图的$κ$连续性的问题在SLE的研究中很自然。为了研究第一个问题，我们考虑了SLE痕迹的自然耦合：对于$κ$的不同值，我们使用相同的布朗运动。很自然地假设有概率，$sle_κ$不断取决于$κ$。很容易表明$ SLE $在Carathéodory的意义上是连续的，但是表明$ SLE $ trace在统一意义上是连续的，这要困难得多。在本说明中，我们表明，对于给定的序列$κ_j\toκ\ in（0，8/3）$，对于几乎每个布朗运动$sle_κ$ traces trace traces trace均匀均匀地收敛。最近，使用不同的方法，Friz，Tran和Yuan也获得了该结果。在我们的分析中，我们提供了一种建设性的方法来研究（0，8/3）$的不同参数$κ\的$SLE_κ$ trace。该参数基于通过布朗运动的分段平方根近似驱动的曲线对SLE曲线近似的新动力学视图。 第二个问题可以在粗糙路径理论的框架中自然回答。使用该理论，我们证明，从$ \sqrtκB_T$驱动的后向后loewner差异方程式的解决方案在$ p $ variation topology中连续$κ$中是连续的，用于所有$κ\
Questions regarding the continuity in $κ$ of the $SLE_κ$ 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, $SLE_κ$ 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 $κ_j\toκ\in (0, 8/3)$, for almost every Brownian motion $SLE_κ$ 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 $SLE_κ$ traces for varying parameter $κ\in (0, 8/3)$. 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 $\sqrtκB_t$ when started away from the origin are continuous in the $p$-variation topology in the parameter $κ$, for all $κ\in \mathbb{R}_+$
