AI 中文总结
研究马尔可夫相互作用粒子耦合随机系统，核心方法围绕正则条件概率，通过对数似然等检测历史条件律分解，贡献是为边界筛选等提供通用语言，还涉及相关能量计算及耦合衡量。
AI 中文摘要
我们考虑分解为外部、边界和内部变量的耦合随机系统，边界变量有时具有传感器和执行器的有向结构。核心问题是历史的条件律何时分解，以及如何通过对数似然、吉拉诺夫测度变换和非平衡统计物理中使用的信息论量来检测这种路径空间陈述。基本对象是路径空间上的正则条件概率。在由固定参考律支配的情况下，边界性质变为拉东 - 尼科迪姆导数的乘法分离，或等效地路径对数似然的加法分离。对于伊藤扩散，通过吉拉诺夫定理计算此对数似然；其期望是在福尔默熵恒等式和薛定谔桥问题的随机控制公式中出现的二次控制能量。当精确分解失败时，剩余耦合通过条件互信息来衡量，即真实边界条件路径律与其条件边缘乘积之间的相对熵。这为边界筛选、路径似然推断、路径律的受控变化以及互信息的热力学值提供了一种通用语言。
英文摘要
We consider coupled stochastic systems decomposed into exterior, boundary, and interior variables, with the boundary variables sometimes carrying the directed structure of a sensor and actuator. The central question is when the conditional law of histories factorises, and how this path space statement is detected by log likelihoods, by Girsanov changes of measure, and by information theoretic quantities used in nonequilibrium statistical physics. The basic object is a regular conditional probability on a path space. Under domination by clamped reference laws, the boundary property becomes multiplicative separation of a Radon--Nikodym derivative, or equivalently additive separation of a path log likelihood. For Itō diffusions this log likelihood is computed by Girsanov's theorem; its expectation is the quadratic control energy appearing in the Föllmer entropy identity and in the stochastic control formulation of Schrödinger bridge problems. When exact factorisation fails, the remaining coupling is measured by conditional mutual information, namely the relative entropy between the true boundary-conditioned path law and the product of its conditional marginals. This gives a common language for boundary screening, path likelihood inference, controlled changes of path law, and the thermodynamic value of mutual information.
Comments35+1 pages
