几何脱钩和多速度Stochastic Klein-Gordon系统中的相色调机制

Geometric Decoupling and the Color--Speed Mechanism in Multispeed Stochastic Klein--Gordon Systems

We study a two-component fractional stochastic Klein--Gordon system on $\mathbb{T}^3$ driven by independent space-time white noises. Our main result is the identification of a new color-speed separation principle for the mixed paracontrolled random operators in the multispeed regime. This principle is created by the interaction between probabilistic cancellation, multispeed dispersive phase separation, and the nonlinear random-operator structure generated by the paracontrolled expansion. At the level of the nonlinear expansion, it reorganizes the singular contraction geometry of the mixed random operators: cross-color contraction channels vanish by independence, while the surviving same-color contractions are forced into speed-separated oscillatory interactions and therefore become finite Volterra-type deterministic multipliers. In this way, the multispeed structure does not merely alter the linear phase geometry; it changes the nonlinear singular interaction mechanism itself and regularizes the most singular contraction channels produced by the quadratic coupling. We prove the almost sure pathwise convergence of the centered fluctuation operators and the convergence of the same-color deterministic contractions, with no singular random-operator centering required in the limiting equation. Our analysis identifies $α= \frac{3}{4}$ as the intrinsic structural threshold of the mixed random-operator algebra. In addition, we construct the baseline stochastic lift and formulate a conditional deterministic closure for the full paracontrolled system. These results show that, in coupled singular dispersive SPDEs, multispeed propagation can act as a geometric regularizer precisely through its effect on the nonlinear random-operator structure.