Gaussian fields and Gaussian sheets with generalized Cauchy covariance structure

Decomposition and limit theorems for a class of self-similar Gaussian processes

We introduce a new class of self-similar Gaussian stochastic processes, where the covariance is defined in terms of a fractional Brownian motion and another Gaussian process. A special case is the solution in time to the fractional-colored stochastic heat equation described in Tudor (2013). We prove that the process can be decomposed into a fractional Brownian motion (with a different parameter than the one that defines the covariance), and a Gaussian process first described in Lei and Nualart (2008). The component processes can be expressed as stochastic integrals with respect to the Brownian…

The persistence exponents of Gaussian random fields connected by the Lamperti transform

The (fractional) Brownian sheet is a simplest example of a Gaussian random field X whose covariance is the tensor product of a finite number (d) of nonnegative correlation functions of self-similar Gaussian processes. Let Y be the homogeneous Gaussian field obtained by applying to X the Lamperti transform, which involves the exponential change of time and the amplitude normalization to have unit variance. Under some assumptions, we prove the existence of the persistence exponents for both fields, X and Y, and find the relation between them. The exponent for X is the limit of ln1/P{X(t)<1, t in…

Gaussian self-similar random fields with distinct stationary properties of their rectangular increments

We describe two classes of Gaussian self-similar random fields: with strictly stationary rectangular increments and with mild stationary rectangular increments. We find explicit spectral and moving average representations for the fields with strictly stationary rectangular increments and characterize fields with mild stationary rectangular increments by the properties of covariance functions of their Lamperti transformations as well as in terms of their spectral densities. We establish that both classes contain not only fractional Brownian sheets and we provide corresponding examples. As a by-…

具有非凸势的微观模型中的Cauchy-Born规则

Cauchy-Born Rule from Microscopic Models with Non-convex Potentials

研究具有非凸相互作用的整数格梯度场模型，通过多尺度重整化群技术，证明了低温小形变下自由能的严格凸性，验证了Cauchy-Born规则，并确定了Gibbs测度Laplace变换的标度极限。

Optimal Convergence Rates and One-Term Edgeworth Expansions for Multidimensional Functionals of Gaussian Fields

We develop techniques for determining the exact asymptotic speed of convergence in the multidimensional normal approximation of smooth functions of Gaussian fields. As a by-product, our findings yield exact limits and often give rise to one-term generalized Edgeworth expansions increasing the speed of convergence. Our main mathematical tools are Malliavin calculus, Stein's method and the Fourth Moment Theorem. This work can be seen as an extension of the results of arXiv:0803.0458 to the multi-dimensional case, with the notable difference that in our framework covariances are allowed to fluctu…

Stochastic Cauchy Initial Value Formulation Of The Heat Equation For Random Field Initial Data: Smoothing, Harnack-Type Bounds And p-Moments

The following stochastic Cauchy initial-value problem is studied for the parabolic heat equation on a domain $ \mathbf{Q}\subset{\mathbf{R}}^{n}$ with random field initial data. \begin{align} &{\square}\widehat{u(x,t)} \equiv \bigg(\frac{\partial}{\partial t}-Δ_{x}\bigg)\widehat{u(x,t)}=0,~x\in\mathbf{Q},t> 0 \end{align} \begin{align} \widehat{u(x,0)}=ϕ(x)+\mathscr{J}(x),~x\in\mathbf{Q},t=0 \end{align} where $ϕ(x)\in C^{\infty}({\mathbf{Q}})$, and $\mathscr{J}(x)$ is a classical Gaussian random scalar field with expectation $\mathbb{E}[\![\mathscr{J}(x)]\!]=0 $ and with a regulated covariance …

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。

高斯过程的高极小值：超调量与极小值点位置

High Minima of Gaussian Processes: Overshoots and Minimizer Locations

研究紧致度量空间上中心高斯过程的高极小值，证明在\(M > u\)条件下，缩放超调量\(u(M - u)\)收敛到指数随机变量，且极小值点条件律的弱子序列极限是最优协方差能量测度，通过多种过程举例说明。

关于平稳高斯场的p-域函数的极限定理

Limit theorems for $p$-domain functionals of stationary Gaussian fields

本文研究了平稳高斯场的p-域函数的中心极限定理和非中心极限定理，探讨了协方差函数可分离或属于Gneiting类时的条件。

高维线性高斯系统样本协方差矩阵的谱统计

Spectral Statistics of the Sample Covariance Matrix for High Dimensional Linear Gaussians

本文研究高维稳定 LTI/LG 系统中 OLS 估计所依赖的样本协方差谱，证明非正规稳定矩阵可使最大特征值指数膨胀并导致估计误差随迭代恶化。