D. Feyel, A. S. Ustunel
Let $(W,H,μ)$ be an abstract Wiener space and $L$ be a probability density of class LlogL. Using the measure transportation of Monge-Kantorovitch, we prove that the kernel of the projection of L on the second Wiener chaos defines an (Hilbert-Schmidt) operator which is lower bounded by another Hilbert-Schmidt operator.
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In this paper, we study the stochastic heat equation with a general multiplicative Gaussian noise that is white in time and colored in space. Both regularity and strict positivity of the densities of the solution have been established. The difficulty, and hence the contribution, of the paper lie in three aspects, which include rough initial conditions, degenerate diffusion coefficient, and weakest possible assumptions on the correlation function of the noise. In particular, our results cover the parabolic Anderson model starting from a Dirac delta initial measure.
Absolute moment inequalities under quadratic-form positivity
本文证明了Zhuang和Hu在Remark 3.1中提出的开放问题，并更一般地考虑了对称联合概率质量函数和联合密度，其关联二次型非负，得到了绝对矩不等式成立的充要条件。
Spectral Criteria for the Asymptotics of Local Functionals of Gaussian Fields and Their Application to Nodal Volumes and Critical Points
本研究提出高斯场局部泛函混沌分量方差正性的谱判据，应用于节体积与临界点分析，推广了现有结果，为第二混沌抵消现象提供新视角。
Spectrally cut-off GFF, regularized $Φ^4$ measure, and reflection positivity
本文论证谱截断高斯自由场不满足空间马尔可夫性质与反射正性，并指出从高斯自由场测度推导正则化 $\Phi^4$ 测度反射正性的朴素方法的困难，填补文献空白。
A remark on elephant random walks via the classical law of the iterated logarithm for self-similar Gaussian processes
本文研究两个独立且记忆参数不同的大象随机游走能否无限次相遇，并利用自相似高斯过程的经典重对数律初等证明其距离的渐近行为。
Let $T$ be a random ergodic pseudometric over $\mathbb R^d$. This setting generalizes the classical \emph{first passage percolation} (FPP) over $\mathbb Z^d$. We provide simple conditions on $T$, the decay of instant one-arms and exponential quasi-independence, that ensure the positivity of its time constants, that is almost surely, the pseudo-distance given by $T$ from the origin is asymptotically a norm. Combining this general result with previously known ones, we prove that The known phase transition for Gaussian percolation in the case of fields with positive correlations with exponentiall…
Let $n,k\geq 1$ and let $G$ be the $n\times n$ random matrix with i.i.d. standard real Gaussian entries. We show that there are constants $c_k,C_k>0$ depending only on $k$ such that the smallest singular value of $G^k$ satisfies $$ c_k\,t\leq {\mathbb P}\big\{s_{\min}(G^k)\leq t^k\,n^{-1/2}\big\}\leq C_k\,t,\quad t\in(0,1], $$ and, furthermore, $$ c_k/t\leq {\mathbb P}\big\{\|G^{-k}\|_{HS}\geq t^k\,n^{1/2}\big\}\leq C_k/t,\quad t\in[1,\infty), $$ where $\|\cdot\|_{HS}$ denotes the Hilbert-Schmidt norm.
Generating Galton-Watson trees using random walks and percolation for the Gaussian free field
本文通过有限随机游走轨迹探索构造Galton-Watson树，结合Dynkin型同构定理，将高斯自由场渗流临界参数$h_*$严格为正的结果推广到任意超临界后代分布和随机电导情形，并证明正相关促进渗流。
For a general class of lattice-spin systems, we prove that an abstract Gaussian concentration bound implies positivity of lower relative entropy density. As a consequence we obtain uniqueness of translation-invariant Gibbs measures from the Gaussian concentration bound in this general setting. This extends earlier results from [Chazottes-Moles-REdig-Ugalde] with a different and very short proof.
In this article we study the connection of fractional Brownian motion, representation theory and reflection positivity in quantum physics. We introduce and study reflection positivity for affine isometric actions of a Lie group on a Hilbert space E and show in particular that fractional Brownian motion for Hurst index 0<H\le 1/2 is reflection positive and leads via reflection positivity to an infinite dimensional Hilbert space if 0<H <1/2. We also study projective invariance of fractional Brownian motion and relate this to the complementary series representations of GL(2,R). We relate this to …
A remark on moment-dependent phase transitions in high-dimensional Gaussian approximations
本文研究了高维高斯近似中维数的临界增长率，探讨了高斯临界值适用于假设检验的界限，并分析了该增长率对观测值矩数量的依赖性。
We prove a new type of Poincaré inequality on abstract Wiener spaces for a family of probability measures which are absolutely continuous with respect to the reference Gaussian measure. This class of probability measures is characterized by the strong positivity (a notion introduced by Nualart and Zakai in [17]) of their Radon-Nikodym densities. Measures of this type do not belong in general to the class of log-concave measures, which are a wide class of measures satisfying the Poincaré inequality (Brascamp and Lieb [2]). Our approach is based on a point-wise identity relating Wick and ordinar…
Gaussian processes occupy one of the leading places in modern statistics and probability theory due to their importance and a wealth of strong results. The common use of Gaussian processes is in connection with problems related to estimation, detection, and many statistical or machine learning models. With the fast development of Gaussian process applications, it is necessary to consolidate the fundamentals of vector-valued stochastic processes, in particular multivariate Gaussian processes, which is the essential theory for many applied problems with multiple correlated responses. In this pap…
In this paper we discuss Sudakov type minoration for the dependent setting. Sudakov minoration is a well known property first proved for centered Gaussian processes which states that for well separated points there is a natural lower bound on the expectation of the supremum of such a process. We generalize this concept for the dependent setting where we consider log concave random variables and then discuss methods of proving the property.
In this note we study a conjecture of Madiman and Wang which predicted that the generalized Gaussian distribution minimizes the Rényi entropy of the sum of independent random variables. Through a variational analysis, we show that the generalized Gaussian fails to be a minimizer for the problem.
Some remarks on Talagrand's convolution conjecture
本文通过鞅方法，利用停止密度鞅和加权伊藤等距估计，为高斯和布尔设定下的Talagrand卷积猜想提供了带显式无维常数的自包含证明。
A remark on Gibbs measures with log-correlated Gaussian fields
本文利用变分公式证明了在 $d$ 维环面上具有对数相关高斯场和四次相互作用的聚焦Gibbs测度的不可正规化性，给出了发散速率，并探讨了三次相互作用及二维Zakharov系统的相关性质。
Beyond the Central Limit: Universality of the Gamma Distribution from Padé-Enhanced Large Deviations
研究揭示伽马分布通过大偏差理论中的Padé近似展开而自然出现，克服了中心极限定理的限制，提供跨学科的普遍性解释。
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