Poisson process Fock space representation, chaos expansion and covariance inequalities

通过传输层次和Malliavin协方差的Wiener混沌中的联合收敛

Joint convergence in Wiener chaos via transport hierarchy and Malliavin covariances

研究多重Wiener-Itô积分序列与高斯极限的联合收敛性，发现联合收敛完全由迭代Malliavin协方差控制，并显式确定极限分布结构。

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…

Decompositions of log-correlated fields with applications

In this article we establish novel decompositions of Gaussian fields taking values in suitable spaces of generalized functions, and then use these decompositions to prove results about Gaussian multiplicative chaos. We prove two decomposition theorems. The first one is a global one and says that if the difference between the covariance kernels of two Gaussian fields, taking values in some Sobolev space, has suitable Sobolev regularity, then these fields differ by a Hölder continuous Gaussian process. Our second decomposition theorem is more specialized and is in the setting of Gaussian fields …

圆周上临界高斯乘性混沌的傅里叶渐近

Fourier asymptotics of critical Gaussian multiplicative chaos on the circle

本文研究圆周上临界高斯乘性混沌的高频傅里叶系数渐近，证明其归一化后稳定收敛到条件稳定噪声，并给出几乎必然积分检验，揭示了空间分布而非仅总质量的极限性质。

Nonlinear Functionals of Hyperbolic Random Waves: the Wiener Chaos Approach

We consider Gaussian random waves on hyperbolic spaces and establish variance asymptotics and central limit theorems for a large class of their integral functionals, both in the high-frequency and large domain limits. Our strategy of proof relies on a fine analysis of Wiener chaos expansions, which in turn requires us to analytically assess the fluctuations of integrals involving mixed moments of covariance kernels. Our results complement several recent findings on non-linear transforms of planar and arithmetic random waves, as well as of random spherical harmonics. In the particular case of 2…

Operators of Gamma white noise analysis

The paper is devoted to the study of Gamma white noise analysis. We define an extended Fock space $\Gama(\Ha)$ over $\Ha=L^2(\R^d, dσ)$, and show how to include the usual Fock space ${\cal F} (\Ha)$ in it as a subspace. We introduce in $\Gama(\Ha)$ operators $a(ξ)=\int_{\R^d} dx ξ(x)a(x)$, $ξ\in S$, with $a(x)=\dig_x+2\dig_x\di_x+1+\di_x +\dig_x\di_x\di_x$, where $\dig_x$ and $\di_x$ are the creation and annihilation operators at $x$. We show that $(a(ξ))_{ξ\in S}$ is a family of commuting selfadjoint operators in $\Gama(\Ha)$ and construct the Fourier transform in generalized joint eigenvecto…

Stochastic analysis of Bernoulli processes

These notes survey some aspects of discrete-time chaotic calculus and its applications, based on the chaos representation property for i.i.d. sequences of random variables. The topics covered include the Clark formula and predictable representation, anticipating calculus, covariance identities and functional inequalities (such as deviation and logarithmic Sobolev inequalities), and an application to option hedging in discrete time.

Stochastic analysis for Poisson processes

This survey is a preliminary version of a chapter of the forthcoming book "Stochastic Analysis for Poisson Point Processes: Malliavin Calculus, Wiener-Itô Chaos Expansions and Stochastic Geometry" edited by Giovanni Peccati and Matthias Reitzner. The paper develops some basic theory for the stochastic analysis of Poisson process on a general $σ$-finite measure space. After giving some fundamental definitions and properties (as the multivariate Mecke equation) the paper presents the Fock space representation of square-integrable functions of a Poisson process in terms of iterated difference ope…

Wiener混沌向量的高斯密度逼近的最优Sobolev速率

Optimal Sobolev Rate for Gaussian Density Approximation of Wiener Chaos Vectors

本文证明Wiener混沌向量高斯逼近在Sobolev空间中的最优密度收敛速率由三阶和四阶累积量决定，并推广至多种距离，方法无需Malliavin非退化条件。

通过混沌展开及相关结果获得几乎必然中心极限定理

Almost sure central limit theorems via chaos expansions and related results

本文通过Wiener混沌展开和Malliavin-Stein方法，在弱条件下建立几乎必然中心极限定理，并推导了二次Wasserstein距离下的定量中心极限定理，解决了Berry随机波模型的Malliavin可微性问题，揭示了Bessel函数q阶矩的渐进行为。

Averaging Gaussian functionals

This paper consists of two parts. In the first part, we focus on the average of a functional over shifted Gaussian homogeneous noise and as the averaging domain covers the whole space, we establish a Breuer-Major type Gaussian fluctuation based on various assumptions on the covariance kernel and/or the spectral measure. Our methodology for the first part begins with the application of Malliavin calculus around Nualart-Peccati's Fourth Moment Theorem, and in addition we apply the Fourier techniques as well as a soft approximation argument based on Bessel functions of first kind. The same method…

Matrix Hermite polynomials, Random determinants and the geometry of Gaussian fields

We study generalized Hermite polynomials with rectangular matrix arguments arising in multivariate statistical analysis and the theory of zonal polynomials. We show that these are well-suited for expressing the Wiener-Ito chaos expansion of functionals of the spectral measure associated with Gaussian matrices. In particular, we obtain the Wiener chaos expansion of Gaussian determinants of the form $\det(XX^T)^{1/2}$ and prove that, in the setting where the rows of $X$ are i.i.d. centred Gaussian vectors with a given covariance matrix, its projection coefficients admit a geometric interpretatio…

Scaling limit of a directed polymer among a Poisson field of independent walks

We consider a directed polymer model in dimension $1+1$, where the disorder is given by the occupation field of a Poisson system of independent random walks on $\mathbb Z$. In a suitable continuum and weak disorder limit, we show that the family of quenched partition functions of the directed polymer converges to the Stratonovich solution of a multiplicative stochastic heat equation (SHE) with a Gaussian noise, whose space-time covariance is given by the heat kernel. In contrast to the case with space-time white noise where the solution of the SHE admits a Wiener-Itô chaos expansion, we establ…