论文标题
关键的布朗乘法混乱
Critical Brownian multiplicative chaos
论文作者
论文摘要
在[JEG20，AHS20，BBK94]中引入的Brownian乘法混乱度量是随机的Borel措施，可以通过将$γ$γ$γ$ times Blanar Brownian运动的平方根进行正式定义。到目前为止，仅研究参数$γ$的亚临界措施。本文考虑了$γ= 2 $的关键案例，使用三个不同的近似程序，这些过程都导致了相同的通用度量。一方面，我们指出了小圆圈当地时代的平方根，并在seneta-heyde归一化以及衍生品的martingale归一化中表现出融合。另一方面，我们将关键措施构建为亚临界措施的限制。这是非高斯关键乘法混乱的第一个例子。 我们受到来自关键高斯乘法混乱的方法的启发，但是存在基本差异，主要的方法是缺乏高斯性，这阻止了卡哈恩（Kahane）的不平等现象的使用，从而导致先验控制。取而代之的是，证明了连续的引理，这使得可以使用随机演算中的工具作为有效的替代品。
Brownian multiplicative chaos measures, introduced in [Jeg20, AHS20, BBK94], are random Borel measures that can be formally defined by exponentiating $γ$ times the square root of the local times of planar Brownian motion. So far, only the subcritical measures where the parameter $γ$ is less than 2 were studied. This article considers the critical case where $γ=2$, using three different approximation procedures which all lead to the same universal measure. On the one hand, we exponentiate the square root of the local times of small circles and show convergence in the Seneta--Heyde normalisation as well as in the derivative martingale normalisation. On the other hand, we construct the critical measure as a limit of subcritical measures. This is the first example of a non-Gaussian critical multiplicative chaos. We are inspired by methods coming from critical Gaussian multiplicative chaos, but there are essential differences, the main one being the lack of Gaussianity which prevents the use of Kahane's inequality and hence a priori controls. Instead, a continuity lemma is proved which makes it possible to use tools from stochastic calculus as an effective substitute.
