The microlocal irregularity of Gaussian noise
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 (1924) 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.
We 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 (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. In particular, the resultant smooth wavefront set 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 embeddings 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.