AI 中文总结
本文开发新框架求解位移 - 波动率场与解析速度，通过伊藤配置映射等方法，得出速度 - 波动率 - 压力系统结构，给出格林 - 久保实现等，形成理论完整、可测试的封闭架构。
AI 中文摘要
大多数随机封闭规定了协方差张量、噪声基或涡粘性场。本文开发了一个不同的框架，其中位移 - 波动率场与解析速度一起求解。起点是单通道伊藤配置映射。局部矩阵对数展开给出不同的物质和空间亨cky增量、它们的二次变差漂移和精确的逐路径体积约束。在恒定密度、一个布朗通道、逐路径等容性以及解析欧拉漂移中无独立鞅的情况下，当保留动量源协变时，物质拉回动量方程被证明是全随机雷诺输运平衡的壳上形式。所得的速度 - 波动率 - 压力系统具有一阶微分 - 代数结构。虚功率确定了应力脉冲的力学类型，并将功与二次协变分开。有限相关前体然后给出已求解位移协方差的格林 - 久保实现。状态依赖性增加了李雅普诺夫噪声诱导的漂移，而动态边界增加了反应功和主动/被动协变兼容性条件。分析还给出了四个极限：非零布朗输运极限不能保留有限的普通未解析动能；仅总能量不能确定熵产生；壁相切限制协方差秩而非随机模式数量；均匀解耦的亥姆霍兹 - 斯托克斯方程只有平凡周期解。结果是一个理论完整、可测试的封闭架构。已开发的湍流统计、对数壁标度和计算流体动力学验证特意留待扩展的流体力学研究。
英文摘要
Most stochastic closures prescribe a covariance tensor, a noise basis or an eddy-viscosity field. This paper develops a different framework in which the displacement-volatility field is solved together with the resolved velocity. The starting point is a one-channel Itô configuration map. A local matrix-logarithm expansion gives distinct material and spatial Hencky increments, their quadratic-variation drifts and the exact pathwise volume constraint. Under constant density, one Brownian channel, pathwise isochoricity and no independent martingale in the resolved Eulerian drift, the material pull-back momentum equation is shown to be the on-shell form of the full stochastic Reynolds transport balance when the momentum-source covariation is retained. The resulting velocity--volatility--pressure system has an index-one differential--algebraic structure. Virtual power fixes the mechanical type of the stress impulse and separates work from quadratic covariation. A finite-correlation precursor then gives a Green--Kubo realisation of the solved displacement covariance. State dependence adds a Lyapunov noise-induced drift, while dynamic boundaries add reaction work and active/passive covariance compatibility conditions. The analysis also gives four limits: a non-zero Brownian transport limit cannot retain finite ordinary unresolved kinetic energy; total energy alone does not fix entropy production; wall tangency limits covariance rank rather than the number of stochastic modes; and a homogeneous decoupled Helmholtz--Stokes equation has only the trivial periodic solution. The result is a theory-complete, testable closure architecture. Developed turbulent statistics, logarithmic wall scaling and computational-fluid-dynamics validation are deliberately left to the expanded fluid-mechanics study.
Comments18 pages, 4 figures
