论文标题
高斯噪声的微局部不规则性
The microlocal irregularity of Gaussian noise
论文作者
论文摘要
对随机傅立叶序列的研究，即具有多项式界定均值和标准偏差的三角函数的线性组合，其系数是独立的（在我们的情况下）随机变量，其历史可以追溯到布朗尼运动的原始结构之一。几何概括（例如，与欧几里得量子场理论具有红外线截止）相关的是对Laplace-Beltrami操作员在任意紧凑的Riemannian riemannian cripold $（M，G）$（M，G）$ Gaussian $，Gaussian noise $ C $ C umcume $ C umcumnian spriportor的征征的随机高斯线性组合的研究。 我将证明，当我们的随机系数是独立的高斯人时，其标准偏差遵守多项式渐近学，并且其含义遵守相应的多项式上限时，随机的随机$ \ mathscr {h}^s $ -waveFront set $ \ perpotatorn set $ \ operatorname {wf}^s（wf}^s（φ） $ \ mathbb {s}^*m $）几乎肯定是空的，或者几乎肯定是$ \ mathbb {s}^*m $的整个，具体取决于$ s \ in \ inthbb {r} $，我们将计算阈值$ s $ s $ s $和Wavefront set的行为。因此，随机$ c^\ infty $ -WaveFront SET $ \ operatorAtorname {wf}（φ）$几乎肯定是整个Cosphere Bundle。 The method of proof is as follows: using Sazonov's theorem and its converse, it suffices to understand which compositions of microlocal cutoffs and inclusions of $L^2$-based fractional order Sobolev spaces are Hilbert-Schmidt (HS), and the answer follows from general facts about the HS-norms of the elements of the pseudodifferential calculus of Kohn and尼伦贝格。
The study of random Fourier series, linear combinations of trigonometric functions whose coefficients are independent (in our case Gaussian) random variables with polynomially bounded means and standard deviations, dates back to Norbert Wiener in one of the original constructions of Brownian motion. A geometric generalization -- relevant e.g.\ to Euclidean quantum field theory with an infrared cutoff -- is the study of random Gaussian linear combinations of the eigenfunctions of the Laplace-Beltrami operator on an arbitrary compact Riemannian manifold $(M,g)$, Gaussian noise $Φ$. I will prove that, when our random coefficients are independent Gaussians whose standard deviations obey polynomial asymptotics and whose means obey a corresponding polynomial upper bound, the resultant random $\mathscr{H}^s$-wavefront set $\operatorname{WF}^s(Φ)$ (defined as a subset of the cosphere bundle $\mathbb{S}^*M$) is either almost surely empty or almost surely the entirety of $\mathbb{S}^*M$, depending on $s \in \mathbb{R}$, and we will compute the threshold $s$ and the behavior of the wavefront set at it. Consequently, the random $C^\infty$-wavefront set $\operatorname{WF}(Φ)$ is almost surely the entirety of the cosphere bundle. The method of proof is as follows: using Sazonov's theorem and its converse, it suffices to understand which compositions of microlocal cutoffs and inclusions of $L^2$-based fractional order Sobolev spaces are Hilbert-Schmidt (HS), and the answer follows from general facts about the HS-norms of the elements of the pseudodifferential calculus of Kohn and Nirenberg.
