Abstract:We prove an abstract operator-level convergence criterion for mixed paracontrolled random operators \[ T^{i;j,k}_{\Lambda}(w) = I_i(w<\Psi_{j,\Lambda})\circ \Psi_{k,\Lambda} \] on \(\mathbb T^d\). These operators are the second-level mixed blocks generated by quadratic paracontrolled expansions, and are singular in the range where the deterministic resonant-product threshold fails. At finite cutoff we separate \[ T^\tau_\Lambda = D^\tau_\Lambda+B^\tau_\Lambda, \] where \(D^\tau_\Lambda\) is the Fourier-diagonal covariance contraction supported on the external diagonal \(n=q\), and \(B^\tau_\Lambda\) is a centered second homogeneous Gaussian chaos with incidence \(n=q+\ell+r\). The centered component is controlled by a finite second-order random tensor estimate, a Banach-valued time lift, and Sobolev--Besov dyadic summation. The deterministic component is treated through a diagonal-resolution interface: retained raw branches or model-determined orbit sums are required only to satisfy scalar Volterra dyadic bounds after the relevant finite cancellations have been formed. The result applies in arbitrary dimension, for finite color/label families, Fourier-diagonal weak covariance, random initial data, colored-in-time Gaussian profiles, and frequency-dependent amplitudes. It abstracts the reusable mixed-operator mechanism underlying concrete color-phase diagonal analyses, while leaving model-specific diagonal classification and scalar Volterra estimates as explicit inputs.
From: Guangqian Zhao [view email]
[v1]
Mon, 15 Jun 2026 12:56:35 UTC (40 KB)
此内容由惯性聚合(RSS阅读器)自动聚合整理，仅供阅读参考。 原文来自 — 版权归原作者所有。