A Singular Stochastic Wave--Klein--Gordon System with Rough Gaussian Forcing and Distinct Propagation Speeds

Abstract

We construct spatially local paracontrolled solutions on $\mathbb{R}^3$ for the stochastic wave--Klein--Gordon system \[ (\partial_t^2-c_{\mathrm{W}}^2\Delta)u=uv+\xi_{\mathrm{W}},\qquad (\partial_t^2-c_{\mathrm{K}}^2\Delta+m^2)v=uv+\xi_{\mathrm{K}}, \] with $c_{\mathrm{W}}\ne c_{\mathrm{K}}$. The independent forcings are white in time and stationary in space, generated by Fourier multipliers of respective orders $\beta_{\mathrm{W}},\beta_{\mathrm{K}}$, and we assume $\beta_*:=\max\{\beta_{\mathrm{W}},\beta_{\mathrm{K}}\}<1/8$. The main analytic step is the construction of the mixed random operators $I_a(w\prec\Psi_b)\circ\Psi_c$. At finite cutoff they decompose into a centered second Gaussian chaos and, when $b=c$, a deterministic covariance contraction. In the latter case the two propagation channels are different; the resulting low--high phase gap gives the diagonal shell bound $N^{-1+2\beta_b}$. The centered terms are controlled through four Schatten flattenings of a continuous-frequency kernel with two physical localizers. Weighted phase-layer estimates also construct the first Picard and cubic stochastic terms. A localized auxiliary system, a finite-cutoff Sobolev bootstrap, and finite propagation yield a compatible family of local solutions with compact-dependent measurable lifetimes. We prove uniqueness in the stated paracontrolled class, local Lipschitz dependence on the enhanced data and Cauchy data, independence of the localization pair, and convergence of the spectral approximations and nonlinear sources. Fixed cutoff profiles converge almost surely along the full sequence; arbitrary admissible cofinal cutoffs converge in probability up to the reference lifetime.