The microlocal irregularity of Gaussian noise

The study of random Fourier series, linear combinations trigonometric functions whose coefficients are independent (in our case Gaussian) variables with polynomially bounded means and standard deviations, dates back to Norbert Wiener in one the original constructions Brownian motion. A geometric generalization -- relevant e.g.\ Euclidean quantum field theory an infrared cutoff is Gaussian eigenfunctions Laplace-Beltrami operator on arbitrary compact Riemannian manifold $(M,g)$, noise $\Phi$. I will prove that, when Gaussians deviations obey polynomial asymptotics a corresponding upper bound, resultant $\mathscr{H}^s$-wavefront set $\operatorname{WF}^s(\Phi)$ (defined as subset cosphere bundle $\mathbb{S}^*M$) either almost surely empty or entirety $\mathbb{S}^*M$, depending $s \in \mathbb{R}$, we compute threshold $s$ behavior wavefront at it. Consequently, $C^\infty$-wavefront $\operatorname{WF}(\Phi)$ bundle. method proof follows: using Sazonov's theorem its converse, it suffices understand which compositions microlocal cutoffs inclusions $L^2$-based fractional order Sobolev spaces Hilbert-Schmidt (HS), answer follows from general facts about HS-norms elements pseudodifferential calculus Kohn Nirenberg.