We develop a probabilistic characterisation of trajectorial expansion rates in non-autonomous stochastic dynamical systems that can be defined over a finite time interval and used for the subsequent uncertainty quantification in Lagrangian (trajectory-based) predictions. These expansion rates are quantified via certain divergences (pre-metrics) between probability measures induced by the laws of the stochastic flow associated with the underlying dynamics. We construct scalar fields of finite-time divergence/expansion rates, show their existence and space-time continuity for general stochastic flows. Combining these divergence rate fields with our 'information inequalities' derived in allows for quantification and mitigation of the uncertainty in path-based observables estimated from simplified models in a way that is amenable to algorithmic implementations, and it can be utilised in information-geometric analysis of statistical estimation and inference, as well as in a data-driven machine/deep learning of coarse-grained models. We also derive a link between the divergence rates and finite-time Lyapunov exponents for probability measures and for path-based observables.
翻译:我们开发了非自主随机动态系统中轨道扩张率的概率特征,可以在有限的时间间隔内加以界定,并用于在Lagrangian(轨道基)预测中随后对不确定性进行量化。这些扩张率通过与基本动态相关的随机流动法则引发的概率测量之间的某些差异(预测)加以量化。我们建造了有限时间差异/扩展率的尺度字段,显示其存在和一般随机流动的时空连续性。将这些差异率字段与我们从我们“信息不平等”中得出的“信息不平等”结合起来,以便能够量化和减轻从简化模型中估计的路径可观察到的不确定性,从而便于算法执行,并且可以用于统计估计和推断的信息测算分析,以及用于数据驱动的机器/对粗差模型的深度学习。我们还从差异率和限时长的Lyapunov前导数据中得出了一种联系,用于概率测量和路径可观测。