Stochastic control and nonequilibrium thermodynamical systems
Stochastic control and nonequilibrium thermodynamical systems
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随机控制和非平衡热力学系统
DOI：
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发表时间：
1989
期刊：
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通讯作者：
M. Pavon
中科院分区：
文献类型：
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作者：
M. Pavon
减去扩散过程的密度的对数被证明是随机控制问题的值函数，其中控制方程随时间向后演化。对于非平衡热力学系统，这提供了一个类似Hamilton-Jacobi的理论，其中的作用是局部熵函数。这个变分原理也可以看作是形式Onsager-Machlup原理的严格版本。对于物理布朗运动的奥恩斯坦-乌伦贝克模型，该原理与路径牛顿定律有关。对于后一种模型，得到了其他几个路径方向的结果，这些结果加强了经典的平均值上的数学结果。特别是，（随机）亥姆霍兹自由能被证明是一个向后下鞅的自然过滤。
Minus the logarithm of the density of a diffusion process is shown to be the value function of a stochastic control problem, where the controlled equation evolves backward in time. For nonequilibrium thermodynamical systems, this provides a Hamilton-Jacobi-like theory, where the action is a local entropy function. This variational principle may also be seen as a rigorous version of the formal Onsager-Machlup principle. For the Ornstein-Uhlenbeck model of physical Brownian motion, the principle is related to a pathwise Newton law. For the latter model, several other pathwise results are derived, which strengthen the classical thermodynamical results on the averages. In particular, the (stochastic) Helmholtz free energy is shown to be a backward submartingale with respect to the natural filtration.