Subcritical Gaussian multiplicative chaos in the Wiener space: construction, moments and volume decay

IF 1.5 1区 数学 Q2 STATISTICS & PROBABILITY
Rodrigo Bazaes, Isabel Lammers, Chiranjib Mukherjee
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引用次数: 0

Abstract

We construct and study properties of an infinite dimensional analog of Kahane’s theory of Gaussian multiplicative chaos (Kahane in Ann Sci Math Quebec 9(2):105-150, 1985). Namely, if \(H_T(\omega )\) is a random field defined w.r.t. space-time white noise \(\dot{B}\) and integrated w.r.t. Brownian paths in \(d\ge 3\), we consider the renormalized exponential \(\mu _{\gamma ,T}\), weighted w.r.t. the Wiener measure \(\mathbb {P}_0(\textrm{d}\omega )\). We construct the almost sure limit \(\mu _\gamma = \lim _{T\rightarrow \infty } \mu _{\gamma ,T}\) in the entire weak disorder (subcritical) regime and call it subcritical GMC on the Wiener space. We show that

$$\begin{aligned} \mu _\gamma \Big \{\omega : \lim _{T\rightarrow \infty } \frac{H_T(\omega )}{T(\phi \star \phi )(0)} \ne \gamma \Big \}=0 \qquad \text{ almost } \text{ surely, } \end{aligned}$$

meaning that \(\mu _\gamma \) is supported almost surely only on \(\gamma \)-thick paths, and consequently, the normalized version is singular w.r.t. the Wiener measure. We then characterize uniquely the limit \(\mu _\gamma \) w.r.t. the mollification scheme \(\phi \) in the sense of Shamov (J Funct Anal 270:3224–3261, 2016) – we show that the law of \(\dot{B}\) under the random rooted measure \(\mathbb Q_{\mu _\gamma }(\textrm{d}\dot{B}\textrm{d}\omega )= \mu _\gamma (\textrm{d}\omega ,\dot{B})P(\textrm{d}\dot{B})\) is the same as the law of the distribution \(f\mapsto \dot{B}(f)+ \gamma \int _0^\infty \int _{\mathbb {R}^d} f(s,y) \phi (\omega _s-y) \textrm{d}s \textrm{d}y\) under \(P \otimes \mathbb {P}_0\). We then determine the fractal properties of the measure around \(\gamma \)-thick paths: \(-C_2 \le \liminf _{\varepsilon \downarrow 0} \varepsilon ^2 \log {\widehat{\mu }}_\gamma (\Vert \omega \Vert< \varepsilon ) \le \limsup _{\varepsilon \downarrow 0}\sup _\eta \varepsilon ^2 \log {\widehat{\mu }}_\gamma (\Vert \omega -\eta \Vert < \varepsilon ) \le -C_1\) w.r.t a weighted norm \(\Vert \cdot \Vert \). Here \(C_1>0\) and \(C_2<\infty \) are the uniform upper (resp. pointwise lower) Hölder exponents which are explicit in the entire weak disorder regime. Moreover, they converge to the scaling exponent of the Wiener measure as the disorder approaches zero. Finally, we establish negative and \(L^p\) (\(p>1\)) moments for the total mass of \(\mu _\gamma \) in the weak disorder regime.

维纳空间中的次临界高斯乘法混沌:构造、矩和体积衰减
我们构建并研究了 Kahane 的高斯乘法混沌理论(Kahane in Ann Sci Math Quebec 9(2):105-150, 1985)的无限维类似物的性质。也就是说,如果\(H_T(\omega )\)是一个在时空白噪声\(\dot{B}\)中定义并在\(dot{B}\)中积分的随机场。在布朗路径中,我们考虑重规范化指数(renormalized exponential \(\mu _{\gamma ,T}\), weighted w.r.t. the Wiener measure \(\mathbb {P}_0(\textrm{d}\omega )\)。我们在整个弱无序(次临界)机制中构造了几乎确定的极限(\mu _\gamma = \lim _{T\rightarrow \infty } \mu _{gamma ,T}/),并将其称为维纳空间上的次临界GMC。我们证明了 $$\begin{aligned}\mu _\gamma \Big \{\omega :\LIM _{T\rightarrow\infty }\(frac{H_T(\omega)}{T(\phi\star\phi )(0)}) (ne\gamma\Big\}=0 ) (qquad\text{ almost }\surely, }\end{aligned}$$meaning that \(\mu _\gamma \) is supported almost surely only on \(\gamma \)-thick路径, and consequently, the normalized version is singular w.r.t. the Wiener measure.然后,我们唯一地描述了极限 \(\mu _\gamma \) w.r.t.在沙莫夫(Shamov)(《函数分析》杂志 270:3224-3261, 2016)--我们证明了在(\mathbb Q_{\mu _\gamma }(\textrm{d}\dot{B}\textrm{d}\omega )= \mu _\gamma (\textrm{d}\omega 、\P(textrm{d}/dot{B})\)与分布定律是一样的 \(f\mapsto \dot{B}(f)+ \gamma \int _0^\infty \int _{\mathbb {R}^d} f(s、y) \phi (\omega _s-y) \textrm{d}s \textrm{d}y\) under \(P \otimes \mathbb {P}_0\).然后我们确定围绕着(gamma)厚路径的度量的分形属性:\(-C_2 \le \liminf _{\varepsilon \downarrow 0} \varepsilon ^2 \log {\widehat{\mu }}\gamma (\Vert \omega \Vert<;\le \limsup _{\varepsilon \downarrow 0}\sup _\eta \varepsilon ^2 \log {widehat\{mu }}_\gamma (\Vert \omega -\eta \Vert < \varepsilon ) \le -C_1\)w.r.t a weighted norm \(\Vert \cdot \Vert \)。这里,(C_1>0\)和(C_2<\infty \)是统一的上部(或者说点状的下部)霍尔德指数,它们在整个弱无序体系中都是显式的。此外,当无序度趋近于零时,它们收敛于维纳度量的缩放指数。最后,我们为弱无序度中\(\mu _\gamma \)的总质量建立了负矩和\(L^p\) (\(p>1\))矩。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
Probability Theory and Related Fields
Probability Theory and Related Fields 数学-统计学与概率论
CiteScore
3.70
自引率
5.00%
发文量
71
审稿时长
6-12 weeks
期刊介绍: Probability Theory and Related Fields publishes research papers in modern probability theory and its various fields of application. Thus, subjects of interest include: mathematical statistical physics, mathematical statistics, mathematical biology, theoretical computer science, and applications of probability theory to other areas of mathematics such as combinatorics, analysis, ergodic theory and geometry. Survey papers on emerging areas of importance may be considered for publication. The main languages of publication are English, French and German.
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