Moments of the Gaussian Chaos

Moments of Gaussian chaoses in Banach spaces

We derive moment and tail estimates for Gaussian chaoses of arbitrary order with values in Banach spaces. We formulate a conjecture regarding two-sided estimates and show that it holds in a certain class of Banach spaces including L_q spaces. As a corollary we obtain two-sided bounds for moments of chaoses with values in L_q spaces based on exponential random variables.

Universal tail profile of Gaussian multiplicative chaos

In this article we study the tail probability of the mass of Gaussian multiplicative chaos. With the novel use of a Tauberian argument and Goldie's implicit renewal theorem, we provide a unified approach to general log-correlated Gaussian fields in arbitrary dimension and derive precise first order asymptotics of the tail probability, resolving a conjecture of Rhodes and Vargas. The leading order is described by a universal constant that captures the generic property of Gaussian multiplicative chaos, and may be seen as the analogue of the Liouville unit volume reflection coefficients in higher…

高斯乘性混沌的逆：一个分部积分公式

Inverse of the Gaussian multiplicative chaos: an integration by parts formula

本文在高斯乘性混沌及其逆的框架下研究分部积分公式的对应版本，推广了高斯过程击中时中的相关结果。

Gaussian multiplicative chaos through the lens of the 2D Gaussian free field

The aim of this review-style paper is to provide a concise, self-contained and unified presentation of the construction and main properties of Gaussian multiplicative chaos (GMC) measures for log-correlated fields in 2D in the subcritical regime. By considering the case of the 2D Gaussian free field, we review convergence, uniqueness and characterisations of the measures; revisit Kahane's convexity inequalities and existence and scaling of moments; discuss the measurability of the underlying field with respect to the GMC measure and present a KPZ relation for scaling exponents.

Universal Gaussian fluctuations on the discrete Poisson chaos

We prove that homogenous sums inside a fixed discrete Poisson chaos are universal with respect to normal approximations. This result parallels some recent findings, in a Gaussian context, by Nourdin, Peccati and Reinert (2010). As a by-product of our analysis, we provide some refinements of the CLTs for random variables on the Poisson space proved by Peccati, Solé, Taqqu and Utzet (2010), and by Peccati and Zheng (2010).

An elementary approach to Gaussian multiplicative chaos

A completely elementary and self-contained proof of convergence of Gaussian multiplicative chaos is given. The argument shows further that the limiting random measure is nontrivial in the entire subcritical phase $(γ< \sqrt{2d})$ and that the limit is universal (i.e., the limiting measure is independent of the regularisation of the underlying field)

High points of a random model of the Riemann-zeta function and Gaussian multiplicative chaos

We study the total mass of high points in a random model for the Riemann-Zeta function. We consider the same model as in [8], [2], and build on the convergence to 'Gaussian' multiplicative chaos proved in [14]. We show that the total mass of points which are a linear order below the maximum divided by their expectation converges almost surely to the Gaussian multiplicative chaos of the approximating Gaussian process times a random function. We use the second moment method together with a branching approximation to establish this convergence.

Multivariate Gaussian approximations on Markov chaoses

We prove a version of the multidimensional Fourth Moment Theorem for chaotic random vectors, in the general context of diffusion Markov generators. In addition to the usual componentwise convergence and unlike the infinite-dimensional Ornstein-Uhlenbeck generator case, another moment-type condition is required to imply joint convergence of of a given sequence of vectors.

Tail universality of critical Gaussian multiplicative chaos

In this article we study the tail probability of the mass of critical Gaussian multiplicative chaos (GMC) associated to a general class of log-correlated Gaussian fields in any dimension, including the Gaussian free field (GFF) in dimension two. More precisely, we derive a fully explicit formula for the leading order asymptotics for the tail probability and demonstrate a new universality phenomenon. Our analysis here shares similar philosophy with the subcritical case but requires a different approach due to complications in the analogous localisation step, and we also employ techniques from r…

Critical Gaussian chaos: convergence and uniqueness in the derivative normalisation

We show that, for general convolution approximations to a large class of log-correlated fields, including the 2d Gaussian free field, the critical chaos measures with derivative normalisation converge to a limiting measure μ'. This limiting measure does not depend on the choice of approximation. Moreover, it is equal to the measure obtained using the Seneta--Heyde renormalisation at criticality, or using a white-noise approximation to the field.

Extremes of Gaussian chaos processes with Trend

Let $\boldsymbol{X}(t)=(X_1(t),\ldots,X_d(t)), t\in [0,S]$ be a Gaussian vector process and let $g(\boldsymbol{x}),\boldsymbol{x}\in\mathbb{R}^d$ be a continuous homogeneous function. In this paper we are concerned with the exact tail asymptotics of the chaos process $g(\boldsymbol{X}(t))+ h(t),t\in [0,S]$ with trend function $h$. Both scenarios $\boldsymbol{X}(t)$ is locally-stationary and $\boldsymbol{X}(t)$ is non-stationary are considered. Important examples include the product of Gaussian processes and chi-processes.

Gaussian Multiplicative Chaos revisited

In this article, we extend the theory of multiplicative chaos for positive definite functions in Rd of the form f(x) = 2 ln+ T|x|+ g(x) where g is a continuous and bounded function. The construction is simpler and more general than the one defined by Kahane in 1985. As main application, we give a rigorous mathematical meaning to the Kolmogorov-Obukhov model of energy dissipation in a turbulent flow.

Integration and stochastic integration in Gaussian multiplicative chaos

We show that for $γ<\sqrt{4/3}$, it is possible to define the Levy area of a planar Brownian motion with the Liouville measure of intermittency parameter $γ$ as the underlying area measure. We also consider the case of smoother curves, and study some properties of the integration map thus defined.

布朗环汤的真实分层场与高斯乘性混沌

The real layering field of Brownian loop soup and the Gaussian multiplicative chaos

本文研究复平面有界单连通域中布朗环汤的分层场，证明其经重整化后收敛到亚临界高斯乘性混沌，主要技术为维纳-伊藤混沌展开，并计算了$n$点函数及其共形协变性和边界行为。

由高斯乘性混沌进行时间变换的过程的收敛性

Convergence of processes time-changed by Gaussian multiplicative chaos

本文在$L^2$区域中证明了由高斯乘性混沌进行时间变换的过程的收敛性，并给出Liouville简单随机游走和Liouville $α$-稳定过程的收敛实例。

Estimates of moments and tails of Gaussian chaoses

We derive two-sided estimates on moments and tails of Gaussian chaoses, that is, random variables of the form $\sum a_{i_1,...,i_d}g_{i_1}... g_{i_d}$, where $g_i$ are i.i.d. ${\mathcal{N}}(0,1)$ r.v.'s. Estimates are exact up to constants depending on $d$ only.

有限个 Wiener 混沌之和的定量中心极限定理及其在 Gaussian 泛函比值中的应用

A quantitative CLT on a finite sum of Wiener chaoses and applications to ratios of Gaussian functionals

本文研究有限 Wiener 混沌和中随机变量部分和的定量中心极限定理，给出总变差距离的新显式界，并用于界定多重随机积分比值与 Gaussian 变量的 Kolmogorov 距离。

Negative moments for Gaussian multiplicative chaos on fractal sets

The objective of this note is to study the probability that the total mass of a sub-critical Gaussian multiplicative chaos (GMC) with arbitrary base measure $σ$ is small. When $σ$ has some continuous density w.r.t Lebesgue measure, a scaling argument shows that the logarithm of the total GMC mass is sub-Gaussian near $-\infty$. However, when $σ$ has no scaling properties, the situation is much less clear. In this paper, we prove that for any base measure $σ$, the total GMC mass has negative moments of all orders.

临界二维随机热流不是高斯乘性混沌

The critical 2d Stochastic Heat Flow is not a Gaussian Multiplicative Chaos

本文证明临界二维随机热流不属于高斯乘性混沌类，通过推导其矩的严格下界实现，该下界具有独立意义。

Multifractal analysis of Gaussian multiplicative chaos and applications

Let $M_γ$ be a subcritical Gaussian multiplicative chaos measure associated with a general log-correlated Gaussian field defined on a bounded domain $D \subset \mathbb{R}^d$, $d \geq 1$. We find an explicit formula for its singularity spectrum by showing that $M_γ$ satisfies almost surely the multifractal formalism, i.e., we prove that its singularity spectrum is almost surely equal to the Legendre-Fenchel transform of its $L^q$-spectrum. Then, applying this result, we compute the lower singularity spectrum of the multifractal random walk and of the Liouville Brownian motion.

Critical Gaussian multiplicative chaos: Convergence of the derivative martingale

In this paper, we study Gaussian multiplicative chaos in the critical case. We show that the so-called derivative martingale, introduced in the context of branching Brownian motions and branching random walks, converges almost surely (in all dimensions) to a random measure with full support. We also show that the limiting measure has no atom. In connection with the derivative martingale, we write explicit conjectures about the glassy phase of log-correlated Gaussian potentials and the relation with the asymptotic expansion of the maximum of log-correlated Gaussian random variables.

Invariance principles for homogeneous sums: Universality of Gaussian Wiener chaos

We compute explicit bounds in the normal and chi-square approximations of multilinear homogenous sums (of arbitrary order) of general centered independent random variables with unit variance. In particular, we show that chaotic random variables enjoy the following form of universality: (a) the normal and chi-square approximations of any homogenous sum can be completely characterized and assessed by first switching to its Wiener chaos counterpart, and (b) the simple upper bounds and convergence criteria available on the Wiener chaos extend almost verbatim to the class of homogeneous sums.

Planar Brownian motion and Gaussian multiplicative chaos

We construct the analogue of Gaussian multiplicative chaos measures for the local times of planar Brownian motion by exponentiating the square root of the local times of small circles. We also consider a flat measure supported on points whose local time is within a constant of the desired thickness level and show a simple relation between the two objects. Our results extend those of Bass, Burdzy and Khoshnevisan and in particular cover the entire $L^1$-phase or subcritical regime. These results allow us to obtain a nondegenerate limit for the appropriately rescaled size of thick points, thereb…

高斯过程签名的维纳混沌展开研究

On the Wiener Chaos Expansion of the Signature of a Gaussian Process

该研究推导了含H∈(1/4,1)的分数布朗运动在内的一类高斯过程签名的维纳混沌分解，得到其期望签名表达式，将已有相关公式推广到H>1/4的一般情形，解决了公开问题，还研究了连续中心化高斯半鞅。

高斯乘性混沌逆的解耦与多点矩

Decoupling and Multipoint moments for the Inverse of the Gaussian multiplicative chaos

本文研究了高斯乘性混沌逆的解耦结构与多点矩，作为扩展随机共形焊接工作至逆Lehto焊接存在性的初步研究，并证明了正实轴上逆同胚的伸缩率属于$L^{1}([0,1]\ imes[0,2])$。

泊松混沌上的伽马近似与泊松-高斯不变性原理

Gamma approximation and Poisson--Gaussian invariance principle on Poisson chaos

该研究在固定泊松混沌上推广伽马近似的相关估计，建立同核泊松与高斯多重积分的不变性原理，为泊松混沌的正态四矩定理提供新证明，并揭示Lindeberg条件的关键作用。

Comparison of moments of Rademacher chaoses

We show that complex hypercontractivity gives better constants than real hypercontractivity in comparison inequalities for (low) moments of Rademacher chaoses (homogeneous polynomials on the discrete cube).

Hölder型估计用于高斯乘法混沌

Hölder type estimates for Gaussian multiplicative chaos

本文研究了GMC比率的右尾行为，通过扩展GMC比率类推导出其上界和下界，利用启发式方法证明最优指数。

The uniqueness of the Gaussian multiplicative chaos revisited

We consider Gaussian multiplicative chaos measures defined in a general setting of metric measure spaces. Uniqueness results are obtained, verifying that different sequences of approximating Gaussian fields lead to the same chaos measure. Specialized to Euclidean spaces, our setup covers both the subcritical chaos and the critical chaos, actually extending to all non-atomic Gaussian chaos measures.

Stein's method on Wiener chaos

We combine Malliavin calculus with Stein's method, in order to derive explicit bounds in the Gaussian and Gamma approximations of random variables in a fixed Wiener chaos of a general Gaussian process. We also prove results concerning random variables admitting a possibly infinite Wiener chaotic decomposition. Our approach generalizes, refines and unifies the central and non-central limit theorems for multiple Wiener-Itô integrals recently proved (in several papers, from 2005 to 2007) by Nourdin, Nualart, Ortiz-Latorre, Peccati and Tudor. We apply our techniques to prove Berry-Esséen bounds in…

Wiener-Chaos Approach to Optimal Prediction

The chaos expansion of a general non-linear function of a Gaussian stationary increment process conditioned on its past realizations is derived. This work combines Wiener chaos expansion approach to study the dynamics of a stochastic system with the classical problem of the prediction of a Gaussian process based on a realization of its past. This is done by constructing a special basis for the Fock space of the Gaussian space generated by the process, such that each basis element is either measurable or independent with respect to the given samples. This property of the basis allows us to deri…

Moment estimates for chaoses generated by symmetric random variables with logarithmically convex tails

We derive two-sided estimates for random multilinear forms (random chaoses) generated by independent symmetric random variables with logarithmically concave tails. Estimates are exact up to multiplicative constants depending only on the order of chaos.

Tail and moment estimates for chaoses generated by symmetric random variables with logarithmically concave tails

We present two-sided estimates of moments and tails of polynomial chaoses of order at most three generated by independent symmetric random variables with log-concave tails as well as for chaoses of arbitrary order generated by independent symmetric exponential variables. The estimates involve only deterministic quantities and are optimal up to constants depending only on the order of the chaos variable.

Two-sided moment estimates for a class of nonnegative chaoses

We derive two-sided bounds for moments of random multilinear forms (random chaoses) with nonnegative coeficients generated by independent nonnegative random variables $X_i$ which satisfy the following condition on the growth of moments: $\lv X_i \rv_{2p} \leq A \lv X_i \rv_p$ for any $i$ and $p\geq 1$. Estimates are deterministic and exact up to multiplicative constants which depend only on the order of chaos and the constant $A$ in the moment assumption.

基于混沌展开的高斯场极限定理及其应用

Limit theorems for Gaussian fields via Chaos Expansions and Applications

该博士论文结合Malliavin演算与Stein方法，研究宇宙学随机超球谐函数非线性泛函定量CLT、金融粗糙波动率分数Ornstein-Uhlenbeck过程问题，还运用大偏差理论分析非自相似随机波动率模型的短时渐近性。

Critical Gaussian Multiplicative Chaos revisited

We present new, short and self-contained proofs of the convergence (with an adequate renormalization) of four different sequences to the critical Gaussian Multiplicative Chaos:(a) the derivative martingale (b) the critical martingale (c) the exponential of the mollified field (d) the subcritical Gaussian Multiplicative Chaos.

Localization of the Gaussian multiplicative chaos in the Wiener space and the stochastic heat equation in strong disorder

We consider a {\it{Gaussian multiplicative chaos}} (GMC) measure on the classical Wiener space driven by a smoothened (Gaussian) space-time white noise. For $d\geq 3$ it was shown in \cite{MSZ16} that for small noise intensity, the total mass of the GMC converges to a strictly positive random variable, while larger disorder strength (i.e., low temperature) forces the total mass to lose uniform integrability, eventually producing a vanishing limit. Inspired by strong localization phenomena for log-correlated Gaussian fields and Gaussian multiplicative chaos in the finite dimensional Euclidean s…

Tail and moment estimates for a class of random chaoses of order two

We derive two-sided bounds for moments and tails of random quadratic forms (random chaoses of order $2$), generated by independent symmetric random variables such that $\lVert X \rVert_{2p} \leq α\lVert X \rVert_p$ for any $p\geq 1$ and some $α\geq 1$. Estimates are deterministic and exact up to some multiplicative constants which depend only on $α$.

Gaussian multiplicative Chaos for symmetric isotropic matrices

Motivated by isotropic fully developed turbulence, we define a theory of symmetric matrix valued isotropic Gaussian multiplicative chaos. Our construction extends the scalar theory developed by J.P. Kahane in 1985.

Four moments theorems on Markov chaos

We obtain quantitative Four Moments Theorems establishing convergence of the laws of elements of a Markov chaos to a Pearson distribution, where the only assumption we make on the Pearson distribution is that it admits four moments. While in general one cannot use moments to establish convergence to a heavy-tailed distributions, we provide a context in which only the first four moments suffices. These results are obtained by proving a general carré du champ bound on the distance between laws of random variables in the domain of a Markov diffusion generator and invariant measures of diffusions.…

基于具有对数凹尾的独立变量的Lq值混沌的矩和尾

Moments and tails of Lq-valued chaoses based on independent variables with log-concave tails

本文推导了任意巴拿赫空间中二阶随机混沌矩的下界，给出了Lq空间中该混沌矩的两个上界（一个需亚高斯假设，一个无需但非最优），并可用于形状参数≥1的威布尔随机变量生成的Lq值混沌的双侧矩估计。

关于欧氏球上具有严格对数协方差的Gaussian multiplicative chaos的小偏差

On small deviations of Gaussian multiplicative chaos with a strictly logarithmic covariance on Euclidean ball

本文证明了在严格对数协方差核的正定性区域内，GMC的小偏差始终为对数正态型，并探讨了其在Laplace变换语境下的显式常数界限与新表示。

超临界高斯乘法混沌的唯一性

Uniqueness of supercritical Gaussian multiplicative chaos

本文研究了超临界高斯乘法混沌测度的唯一性，证明其收敛到非平凡极限，且该极限仅由正则化选择决定。

通过高斯乘性混沌构造的随机解析函数

Random analytic functions via Gaussian multiplicative chaos

本研究以高斯乘性测度为Clark测度定义单位圆盘上的随机解析函数$\varphi$，证明其几乎必然为Blaschke乘积，并给出零点密度的精确估计。

Gaussian Multiplicative Chaos for Gaussian Orthogonal and Symplectic Ensembles

We study the characteristic polynomials of both the Gaussian Orthogonal and Symplectic Ensembles. We show that for both ensembles, powers of the absolute value of the characteristic polynomials converge in law to Gaussian multiplicative chaos measures after normalization for sufficiently small real powers. The main tool is a new asymptotic relation between the fractional moments of the absolute characteristic polynomials of Gaussian Orthogonal, Unitary, and Symplectic Ensembles.

二阶高斯混沌的算子范数Sudakov下界

Operator-norm Sudakov minoration for Gaussian chaos of order two

本文证明二阶高斯混沌的算子范数Sudakov下界，通过自适应高斯实验和凸分离方法，给出分离矩阵族期望上确界的下界估计。

高斯乘性混沌测度的体/边界商 II：精确尺度不变情形下体高斯乘性混沌测度的尾部轮廓

Bulk/boundary quotients of Gaussian multiplicative chaos measures II: Tail profile of bulk Gaussian multiplicative chaos measures in the exact scale-invariant case

本文研究精确尺度不变核下边界具有均匀奇异性的高斯乘性混沌测度的右尾轮廓，通过将局部化技巧应用于边界上的辅助测度，推广了 Rhodes-Vargas 和 Wong 的平坦背景几何结果，并用体/边界商的变体刻画尾部轮廓系数。

A Theory of Intermittency Renormalization of Gaussian Multiplicative Chaos Measures

A theory of intermittency differentiation is developed for a general class of Gaussian Multiplicative Chaos measures including the measure of Bacry and Muzy on the interval and circle as special cases. An exact, non-local functional equation is derived for the derivative of a general functional of the total mass of the measure with respect to intermittency. The formal solution is given in the form of an intermittency expansion and proved to be a renormalized expansion in the centered moments of the total mass of the measure. The full intermittency expansion of the Mellin transform of the total…

Chaos of a Markov operator and the fourth moment condition

We analyze from the viewpoint of an abstract Markov operator recent results by Nualart and Peccati, and Nourdin and Peccati, on the fourth moment as a condition on a Wiener chaos to have a distribution close to Gaussian. In particular, we are led to introduce a notion of chaos associated to a Markov operator through its iterated gradients and present conditions on the (pure) point spectrum for a sequence of chaos eigenfunctions to converge to a Gaussian distribution. Convergence to gamma distributions may be examined similarly.

The conditional Gaussian multiplicative chaos structure underlying a critical continuum random polymer model on a diamond fractal

We discuss a Gaussian multiplicative chaos (GMC) structure underlying a family of random measures $\mathbf{M}_r$, indexed by $r\in\mathbb{R}$, on a space $Γ$ of directed pathways crossing a diamond fractal with Hausdorff dimension two. The laws of these random continuum path measures arise in a critical weak-disorder limiting regime for discrete directed polymers on disordered hierarchical graphs. For the analogous subcritical continuum polymer model in which the diamond fractal has Hausdorff dimension less than two, the random path measures can be constructed as subcritical GMCs through coupl…

Glassy phase and freezing of log-correlated Gaussian potentials

In this paper, we consider the Gibbs measure associated to a logarithmically correlated random potential (including two dimensional free fields) at low temperature. We prove that the energy landscape freezes and enters in the so-called glassy phase. The limiting Gibbs weights are integrated atomic random measures with random intensity expressed in terms of the critical Gaussian multiplicative chaos. This could be seen as a first rigorous step in the renormalization theory of super-critical Gaussian multiplicative chaos.

A four moments theorem for Gamma limits on a Poisson chaos

This paper deals with sequences of random variables belonging to a fixed chaos of order $q$ generated by a Poisson random measure on a Polish space. The problem is investigated whether convergence of the third and fourth moment of such a suitably normalized sequence to the third and fourth moment of a centred Gamma law implies convergence in distribution of the involved random variables. A positive answer is obtained for $q=2$ and $q=4$. The proof of this four moments theorem is based on a number of new estimates for contraction norms. Applications concern homogeneous sums and $U$-statistics o…

Gaussian multiplicative chaos and KPZ duality

This paper is concerned with the construction of atomic Gaussian multiplicative chaos and the KPZ formula in Liouville quantum gravity. On the first hand, we construct purely atomic random measures corresponding to values of the parameter $γ^2$ beyond the transition phase (i.e. $γ^2>2d$) and check the duality relation with sub-critical Gaussian multiplicative chaos. On the other hand, we give a simplified proof of the classical KPZ formula as well as the dual KPZ formula for atomic Gaussian multiplicative chaos. In particular, this framework allows to construct singular Liouville measures and …

Cumulants on Wiener chaos: moderate deviations and the fourth moment theorem

A moderate deviation principle as well as moderate and large deviation inequalities for a sequence of elements living inside a fixed Wiener chaos associated with an isonormal Gaussian process are shown. The conditions under which the results are derived coincide with those of the celebrated fourth moment theorem of Nualart and Peccati. The proofs rely on sharp estimates for cumulants. As applications, explosive integrals of a Brownian sheet, a discretized version of the quadratic variation of a fractional Brownian motion and the sample bispectrum of a spherical Gaussian random field are consid…