Color--Phase Separation for Mixed Random Operators in Two-Speed Stochastic Klein--Gordon Systems

Abstract

We study a two-component stochastic fractional Klein--Gordon system on the three-dimensional torus with fractional dispersion exponent \(\alpha\), distinct propagation speeds, and a pure cross nonlinearity. The singular mixed terms combine a low--high paraproduct inside a Duhamel integral with an outer resonant product. Their structure is governed by two index pairs: the color pair determines the possible Gaussian contraction, whereas the propagator--input pair determines the Duhamel phase. In every same-color block generated by the interaction graph, the outer propagator has a different speed. The contracted term is therefore a Fourier-diagonal Volterra multiplier, and integration by parts in time yields a factor of order \(N^{-\alpha}\) at frequency scale \(N\). For the centered remainder we give a finite Hilbert-space kernel normal form that preserves the Fourier incidence relation and reduces the operator estimate to oriented Gaussian-tensor flattenings. For independent noises we construct the enhanced stochastic data, prove local well-posedness in the prescribed enhanced paracontrolled/Galerkin class, and establish pathwise convergence of the Fourier--Galerkin equations for \(12/13<\alpha\le1\). We also prove a logically separate perturbative extension to Fourier-diagonal color covariances with frequency-decaying off-diagonal entries.