After coarse-graining a complex system, the dynamics of its macro-state may exhibit more pronounced causal effects than those of its micro-state. This phenomenon, known as causal emergence, is quantified by the indicator of effective information. However, two challenges confront this theory: the absence of well-developed frameworks in continuous stochastic dynamical systems and the reliance on coarse-graining methodologies. In this study, we introduce an exact theoretic framework for causal emergence within linear stochastic iteration systems featuring continuous state spaces and Gaussian noise. Building upon this foundation, we derive an analytical expression for effective information across general dynamics and identify optimal linear coarse-graining strategies that maximize the degree of causal emergence when the dimension averaged uncertainty eliminated by coarse-graining has an upper bound. Our investigation reveals that the maximal causal emergence and the optimal coarse-graining methods are primarily determined by the principal eigenvalues and eigenvectors of the dynamic system's parameter matrix, with the latter not being unique. To validate our propositions, we apply our analytical models to three simplified physical systems, comparing the outcomes with numerical simulations, and consistently achieve congruent results.

翻译：对复杂系统进行粗粒化后，其宏观态动力学可能比微观态展现出更显著的因果效应。这种被称为因果涌现的现象，通过有效信息指标来量化。然而，该理论面临两大挑战：缺乏连续随机动力系统的成熟框架，以及对粗粒化方法的依赖。本研究针对具有连续状态空间和高斯噪声的线性随机迭代系统，提出了因果涌现的精确理论框架。在此基础上，我们推导了一般动力学中有效信息的解析表达式，并确定了在粗粒化消除的平均不确定性存在上界时，能够最大化因果涌现程度的最优线性粗粒化策略。研究表明，最大因果涌现程度与最优粗粒化方法主要由动力学系统参数矩阵的主特征值和特征向量决定，且后者并非唯一。为验证理论命题，我们将分析模型应用于三个简化物理系统，将结果与数值模拟进行对比，均获得了一致结论。