论文标题
Schramm-loewner进化的规律性：最新的变化和连续性模量
Regularity of the Schramm-Loewner evolution: Up-to-constant variation and modulus of continuity
论文作者
论文摘要
我们发现，对于Schramm-loewner Evolution（SLE）的规律性，我们发现了最佳（最高）边界：变化规律性，连续性模量和迭代对数定律。对于后两者，我们认为SLE具有其自然参数化。更准确地说，用$ d \表示（0,2] $曲线的尺寸，我们显示以下内容。 1。最佳$ψ$ - 变量为$ψ（x）= x^d（\ log \ log \ log x^{ - 1}）^{ - （d-1）} $，因为$η$是A.S。此$ψ$的有限$ψ$变量，而不会衰减的任何功能，因为$ x \ downrow 0 $。 2。连续性的最佳模量为$ω（s）= c \，s^{1/d}（\ log s^{ - 1}）^{1-1/d} $，即对于某些随机$ c> 0 $ \leΩ（t-s）$ a.s.，而对于任何功能$ω$衰减的速度都不含为$ s \ downarrow 0 $。 3。$ \ limsup_ {t \ downarrow 0} |η（t）| \，\ big（t^{1/d}（\ log \ log \ log \ log \ log t^{ - 1}）^{1-1/d} \ big）等于$（0，\ infty）$中的确定性常数。 我们还表明，SLE的自然参数是由$ψ$变量的细网限制给出的。作为证据的一部分，我们表明，每个随机过程都满足特定力矩条件的范围都会达到一定的变化规律性。
We find optimal (up to constant) bounds for the following measures for the regularity of the Schramm-Loewner evolution (SLE): variation regularity, modulus of continuity, and law of the iterated logarithm. For the latter two we consider the SLE with its natural parametrisation. More precisely, denoting by $d\in(0,2]$ the dimension of the curve, we show the following. 1. The optimal $ψ$-variation is $ψ(x)=x^d(\log\log x^{-1})^{-(d-1)}$ in the sense that $η$ is a.s. of finite $ψ$-variation for this $ψ$ and not for any function decaying more slowly as $x \downarrow 0$. 2. The optimal modulus of continuity is $ω(s) = c\,s^{1/d}(\log s^{-1})^{1-1/d}$, i.e. for some random $c>0$ we have $|η(t)-η(s)| \le ω(t-s)$ a.s., while this does not hold for any function $ω$ decaying faster as $s \downarrow 0$. 3. $\limsup_{t\downarrow 0} |η(t)|\,\big(t^{1/d}(\log\log t^{-1})^{1-1/d}\big)^{-1}$ is a.s. equal to a deterministic constant in $(0,\infty)$. We also show that the natural parametrisation of SLE is given by the fine mesh limit of the $ψ$-variation. As part of our proof, we show that every stochastic process whose increments satisfy a particular moment condition attains a certain variation regularity.
