Stochastic Komatu–Loewner Evolutions
Stochastic Komatu–Loewner Evolutions
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DOI：
10.1142/13038
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发表时间：
2022-05
期刊：
影响因子：
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通讯作者：
Zhen-Qing Chen;M. Fukushima;Takuya Murayama
中科院分区：
文献类型：
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作者：
Zhen-Qing Chen;M. Fukushima;Takuya Murayama
Loewner 方程是共角映射的微分方程，可用于描述一族简单连接的平面域的演化。它是由 C. Loewner 于 1923 年在他关于比伯巴赫猜想的著作中引入的。 Oded Schramm 在 2000 年观察并猜想，统计物理中许多二维晶格模型的标度极限可以用布朗运动为驱动函数的 Loewner 演化来描述。其中许多猜想后来在 G. Lawler、O. Schramm 和 W. Werner 以及 S. Smirnov 的一系列联合工作中得到证实。另一方面，Y. Komatu 在 1950 年将 Loewner 方程扩展到圆缝环，但是是左导数意义上的。本系列讲座的目的是调查 Komatu-Loewner 演化及其在正则狭缝域中的随机对应物研究的一些最新进展，重点是概率方法。
Loewner equation is a differential equation for conformal mappings that can be used to describe evolution of a family of simply connected planar domains. It was introduced by C. Loewner in 1923 in his work on the Bieberbach conjecture. Oded Schramm observed and conjectured in 2000 that scaling limit of many two-dimensional lattice models in statistical physics can be described by Loewner evolutions with Brownian motions as the driving function. Many of these conjectures are latter confirmed in a series of joint work by G. Lawler, O. Schramm and W. Werner and by S. Smirnov. On the other hand, Y. Komatu extended Loewner equation to circularly slit annuli in 1950 but in the left derivative sense. The aim of this series of lectures is to survey some recent progress in the study of Komatu-Loewner evolutions and its stochastic counterpart in the canonical slit domains, with emphasis on probabilistic methods.