Loewner evolution driven by complex Brownian motion

IF 2.1 1区 数学 Q1 STATISTICS & PROBABILITY
Ewain Gwynne, Joshua Pfeffer, Minjae Park
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引用次数: 1

Abstract

We study the Loewner evolution whose driving function is Wt=Bt1+iBt2, where (B1,B2) is a pair of Brownian motions with a given covariance matrix. This model can be thought of as a generalization of Schramm–Loewner evolution (SLE) with complex parameter values. We show that our Loewner evolutions behave very differently from ordinary SLE. For example, if neither B1 nor B2 is identically equal to zero, then the set of points disconnected from ∞ by the Loewner hull has nonempty interior at each time. We also show that our model exhibits three phases analogous to the phases of SLE: a phase where the hulls have zero Lebesgue measure, a phase where points are swallowed but not hit by the hulls and a phase where the hulls are space-filling. The phase boundaries are expressed in terms of the signs of explicit integrals. These boundaries have a simple closed form when the correlation of the two Brownian motions is zero.
由复杂布朗运动驱动的低级进化
我们研究了驱动函数为Wt=Bt1+iBt2的Loewner演化,其中(B1,B2)是一对具有给定协方差矩阵的布朗运动。该模型可以被认为是具有复杂参数值的Schramm-Loewner演化(SLE)的推广。我们的研究表明,我们的下层进化与普通SLE的表现非常不同。例如,如果B1和B2都不等于零,那么被Loewner船体从∞断开的点集每次都具有非空的内部。我们还表明,我们的模型显示出与SLE相类似的三个阶段:一个阶段,船体没有勒贝格测量值,一个阶段,点被吞噬但不被船体击中,一个阶段,船体填充空间。相边界用显式积分的符号表示。当两个布朗运动的相关性为零时,这些边界具有简单的封闭形式。
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来源期刊
Annals of Probability
Annals of Probability 数学-统计学与概率论
CiteScore
4.60
自引率
8.70%
发文量
61
审稿时长
6-12 weeks
期刊介绍: The Annals of Probability publishes research papers in modern probability theory, its relations to other areas of mathematics, and its applications in the physical and biological sciences. Emphasis is on importance, interest, and originality – formal novelty and correctness are not sufficient for publication. The Annals will also publish authoritative review papers and surveys of areas in vigorous development.
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