随机游动和分形曲线驱动的迭代Schwarz-Christoffel变换
Iterative Schwarz-Christoffel Transformations Driven by Random Walks and Fractal Curves
摘要：
随机Loewner演化（Stochastic Loewner evolution，SLE）是由一维布朗运动（BM）驱动的微分方程，其解在复平面$\H$上给出一个保角变换的随机过程.作为变换图像的演化边界，生成了一条随机曲线（SLE曲线），该曲线从原点开始，随着时间的推移，沿着$\H$向无穷远处运行。SLE曲线提供了各种统计合奏的重要分形曲线，如果我们改变的扩散常数的驱动BM。本文研究了Schwarz-Christoffel变换（SCT），它是从$\H$到从原点开始有狭缝的区域$\H$的共形映射.我们准备了一个二项式系统的SCT，其中一个产生一个狭缝在$\H$与角度$απ$从实轴的正方向，另一个与角度$(1-α) π$。引入一个参数$κ>0$来控制$α$的值和狭缝的长度。在一维随机游走（这是一个二项式随机过程）的驱动下，执行SCT的随机迭代。在$\H$中，通过用直线插值缝尖，得到一条随机路径，称为迭代SCT（Iterative SCT，ISCT）路径.众所周知，当随机游动的步数$N$趋于无穷大时，每条随机游动路径除以$\sqrt{N}$均收敛于一条布朗曲线.然后，我们期望由$\sqrt{N}$划分的ISCT路径（重新缩放的ISCT路径）收敛到$N \to \infty$中的SLE曲线。我们的数值研究表明，对于足够大的$N$，重新标度的ISCT路径将具有相同的统计特性的SLE曲线，支持我们的期望。
英文摘要：
Stochastic Loewner evolution (SLE) is a differential equation driven by a one-dimensional Brownian motion (BM), whose solution gives a stochastic process of conformal transformation on the upper half complex-plane $\H$. As an evolutionary boundary of image of the transformation, a random curve (the SLE curve) is generated, which is starting from the origin and running in $\H$ toward the infinity as time is going. The SLE curves provides a variety of statistical ensembles of important fractal curves, if we change the diffusion constant of the driving BM. In the present paper, we consider the Schwarz-Christoffel transformation (SCT), which is a conformal map from $\H$ to the region $\H$ with a slit starting from the origin. We prepare a binomial system of SCTs, one of which generates a slit in $\H$ with an angle $απ$ from the positive direction of the real axis, and the other of which with an angle $(1-α) π$. One parameter $κ>0$ is introduced to control the value of $α$ and the length of slit. Driven by a one-dimensional random walk, which is a binomial stochastic process, a random iteration of SCTs is performed. By interpolating tips of slits by straight lines, we have a random path in $\H$, which we call an Iterative SCT (ISCT) path. It is well-known that, as the number of steps $N$ of random walk goes infinity, each path of random walk divided by $\sqrt{N}$ converges to a Brownian curve. Then we expect that the ISCT paths divided by $\sqrt{N}$ (the rescaled ISCT paths) converge to the SLE curves in $N \to \infty$. Our numerical study implies that, for sufficiently large $N$, the rescaled ISCT paths will have the same statistical properties as the SLE curves have, supporting our expectation.
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