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

Abstract

We study a two-component stochastic Klein--Gordon system on \(\mathbb T^3\) with fixed distinct speeds and pure cross interaction \(u_1u_2\). The mixed paracontrolled operators \[ I_i(w<Ψ_j)\circ Ψ_k \] are organized by color--phase separation: the pair \((j,k)\) determines the Wick or covariance contraction, while the pair \((i,j)\) determines the Duhamel--source phase gap. In the pure-cross graph, same-color contractions occur only in different-phase channels and become Fourier-diagonal Volterra multipliers; the remaining centered kernels are controlled as operator-valued second Gaussian chaoses by row/column tensor estimates. This yields a stochastic enhanced-data construction, a local paracontrolled solution map, and canonical Galerkin convergence. The result covers diagonal independent noises and Fourier-diagonal weak covariance. For \(12/13<α<1\) the deterministic map uses a fractional Klein--Gordon Strichartz estimate proved here, while the endpoint \(α=1\) uses the classical conic wave/Klein--Gordon package.