Random Tensor Estimates and Deterministic Diagonal Resolutions for Mixed Paracontrolled Operators

Abstract

We prove an operator-level convergence theorem for the mixed paracontrolled blocks \[ T^{i;j,k}_{\Lambda}(w) =I_i\bigl(w<\Psi_{j,\Lambda}\bigr)\circ\Psi_{k,\Lambda} \] on \(\mathbb T^d\), under Fourier-diagonal Gaussian covariance. At finite Galerkin cutoff, the Wick decomposition is \[ T^\tau_\Lambda=D^\tau_\Lambda+B^\tau_\Lambda, \qquad \tau=(i;j,k), \] where \(D^\tau_\Lambda\) is a deterministic Volterra multiplier on the external diagonal \(n=q\), while \(B^\tau_\Lambda\) is a centered second homogeneous Gaussian chaos with incidence \(n=q+\ell+r\). The centered coefficient has two Gaussian frequency legs and one input-output pair. Write \(\Delta_T=\{(t,s):0\le s\le t\le T\}\). An iterated rectangular non-commutative Khintchine argument, followed by the four associated oriented flattenings, yields \[ \bigl\|B^\tau_{\Lambda,N,Q,M}\bigr\|_ {L^p\bigl(\Omega;C(\Delta_T;\mathcal L(\ell_q^2,\ell_n^2))\bigr)} \lesssim_{p,\varepsilon} N^{d/2-\Gamma_\tau+\varepsilon} \bigl(M^{d/2}+Q^{d/2}\bigr), \qquad \Gamma_\tau=\lambda_i+\alpha_j+\alpha_k. \] Under the resulting strict Sobolev--Besov summability conditions, the centered cutoffs converge in \(L^p(\Omega)\) and almost surely in operator norm. The covariance contraction is kept separate from the tensor estimate: retained raw branches, or prescribed finite combinations of branches, enter through a scalar Volterra criterion imposed after the relevant cancellation has been formed. We give a frequency-envelope variant and verify the diagonal criterion for distinct-speed wave and Klein--Gordon contractions.