矩阵噪声雅可比在随机微积分推断和最优路径中
Matrix-noise Jacobians in stochastic-calculus inference and optimal paths
AI总结：
研究揭示了矩阵噪声雅可比在随机动力学中的影响，通过有限步路径似然框架展示其在短时展开中的作用，并通过实验验证了其在优化路径中的可测量变化。
摘要：
乘性噪声使随机动力学依赖于白噪声极限的解释方式。在多维系统中，矩阵值噪声幅值σ(x)的依赖性包含局部雅可比贡献，这在标量示例中不存在。本文提出了θ离散化扩散的有限步路径似然框架，并展示其短时展开分离出标量J_σ=∂_jσ_{ik}∂_iσ_{jk}-(∂_iσ_{ik})(∂_lσ_{lk})。对于指定的噪声幅值表示σ，该量在一维、标量各向同性和严格对角情况下消失，但在状态依赖噪声方向混合不同组件时可存活。随后通过配对比较验证其后果，固定漂移、扩散矩阵、插值点和高斯增量项。在模型A中，移除仅非对角行列式贡献导致拟合随机性描述的偏移，当J_σ=0时消失。在模型B中，移除相应状态依赖作用项改变稳定优化过渡路径。这些结果表明，短时路径测度中真正的矩阵噪声部分可存活于简单设置中熟悉的标量抵消，并在拟合随机性描述和奥萨格-马赫鲁普路径中产生可测量变化。
英文摘要：
Multiplicative noise makes stochastic dynamics depend on how the white-noise limit is interpreted. In multidimensional systems with matrix-valued noise amplitudes $σ(x)$, this dependence includes a local Jacobian contribution that is absent from the scalar examples most often used to build intuition. We formulate a finite-step path-likelihood framework for $θ$-discretized diffusions and show that its short-time expansion isolates the scalar $J_σ=\partial_jσ_{ik}\partial_iσ_{jk}-(\partial_iσ_{ik})(\partial_lσ_{lk})$. For a specified noise-amplitude representation $σ$, this quantity vanishes in one-dimensional, scalar-isotropic, and strictly diagonal cases, but can survive when state-dependent noise directions mix different components. We then test its consequences using paired comparisons that hold the drift, diffusion matrix, interpolation point, and Gaussian increment term fixed. In Model A, removing only the off-diagonal determinant contribution produces a shift in the fitted stochastic prescription that vanishes when $J_σ=0$. In Model B, removing the corresponding state-dependent action term changes a stable optimized transition path. These results show that a genuinely matrix-noise part of the short-time path measure can survive the scalar cancellations familiar from simpler settings and produce measurable changes in fitted stochastic prescriptions and Onsager--Machlup paths.
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