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

TL;DR

This paper investigates a two-speed stochastic Klein-Gordon system on a7^3, revealing a color-phase separation mechanism that influences Wick contractions and covariance structures, leading to a local paracontrolled solution.

Contribution

It introduces a novel color-phase separation framework for mixed random operators in a two-speed stochastic Klein-Gordon system, with new operator-valued Gaussian chaos analysis and solution construction methods.

Findings

Color labels determine Wick contractions and covariance blocks.

Phase labels record the Duhamel-source phase difference.

A new Hilbert-kernel normal form and tensor estimates enable source bounds.

Abstract

We study a two-component stochastic Klein-Gordon system on \(\mathbb T^3\) with fixed distinct speeds, pure cross interaction \(u_1u_2\), and diagonal independent space-time white noises. The mixed paracontrolled random operators exhibit a color-phase separation mechanism: color labels determine Wick contractions and covariance blocks, while phase labels record the Duhamel-source phase difference between the outer propagator and the stochastic high-frequency leg. In the present two-speed model this phase difference is produced by the low-high bound \(|\omega_i(\ell+q)-\omega_j(\ell)|\gtrsim N^\alpha\) for \(i\ne j\). Same-color contractions therefore occur only in different-phase Duhamel channels and become finite Fourier-diagonal Volterra multipliers; same-Duhamel-source blocks are cross-color and are centered by independence. After subtracting the Volterra diagonal, the remaining mixed kernels are operator-valued second homogeneous Gaussian chaoses. A finite Hilbert-kernel normal form preserves the incidence \(n=q+\ell+r\), and a row/column tensor estimate with profile \[ N^{3/2-3\alpha+}(M^{3/2+}+Q^{3/2+})Q^{-\sigma} \] yields both Sobolev and \(L^1_TB^{\sigma-\alpha}_{2,\infty}\) source bounds. Combining the stochastic enhanced-data construction with the deterministic solution map gives a local paracontrolled solution and canonical Galerkin convergence. For \(12/13<\alpha<1\) the deterministic map uses an internal fractional Klein-Gordon Strichartz theorem proved here; for \(\alpha=1\) it uses the classical conic wave/Klein-Gordon package. The theorem is restricted to diagonal independent colors, fixed distinct speeds, and pure cross interaction. Weakly correlated colors are recorded only at the finite-cutoff algebraic level; no correlated-color local theorem is claimed.