具有粗糙高斯强迫和不同传播速度的奇异随机波动-克莱因-戈登系统
A Singular Stochastic Wave--Klein--Gordon System with Rough Gaussian Forcing and Distinct Propagation Speeds
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中文总结 AI 辅助
研究具有不同传播速度和粗糙高斯强迫的随机波动-克莱因-戈登系统，通过构造混合随机算子等方法，在\(\mathbb{R}^3\)上得到空间局部仿控制解，证明了唯一性、依赖性等性质及相关收敛性。
中文摘要 AI 辅助
我们在\(\mathbb{R}^3\)上构造了随机波动-克莱因-戈登系统\[ (\partial_t^2 - c_{\mathrm{W}}^2\Delta)u = uv + \xi_{\mathrm{W}},\qquad (\partial_t^2 - c_{\mathrm{K}}^2\Delta + m^2)v = uv + \xi_{\mathrm{K}} \]的空间局部仿控制解，其中\(c_{\mathrm{W}}\neq c_{\mathrm{K}}\)。独立强迫项在时间上是白噪声且在空间上是平稳的，由各自阶数为\(\beta_{\mathrm{W}},\beta_{\mathrm{K}}\)的傅里叶乘子生成，且假设\(\beta_* := \max\{\beta_{\mathrm{W}},\beta_{\mathrm{K}}\} < 1/8\)。主要分析步骤是构造混合随机算子\(I_a(w\prec\Psi_b)\circ\Psi_c\)。在有限截断时它们分解为中心二阶高斯混沌，当\(b = c\)时为确定性协方差收缩。通过具有两个物理局部化器的连续频率核的四个施坦因扁平化来控制中心项。加权相位层估计还构造了第一个皮卡和三次随机项。一个局部辅助系统、有限截断索伯列夫引导和有限传播产生了一族具有紧依赖可测寿命的兼容局部解。我们证明了在所述仿控制类中的唯一性、对增强数据和柯西数据的局部利普希茨依赖性、对局部化对的独立性以及谱逼近和非线性源的收敛性。固定截断轮廓几乎必然沿整个序列收敛；任意可允许的共尾截断在参考寿命之前依概率收敛。
英文摘要
We construct spatially local paracontrolled solutions on $\mathbb{R}^3$ for the stochastic wave--Klein--Gordon system \[ (\partial_t^2-c_{\mathrm{W}}^2Δ)u=uv+ξ_{\mathrm{W}},\qquad (\partial_t^2-c_{\mathrm{K}}^2Δ+m^2)v=uv+ξ_{\mathrm{K}}, \] 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 $β_{\mathrm{W}},β_{\mathrm{K}}$, and we assume $β_*:=\max\{β_{\mathrm{W}},β_{\mathrm{K}}\}<1/8$. The main analytic step is the construction of the mixed random operators $I_a(w\precΨ_b)\circΨ_c$. 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 $N^{-1+2β_b}$. 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.
