利用“量子力学”算子构造随机混合路径积分
Construction of stochastic hybrid path integrals using "quantum-mechanical'' operators
摘要：
随机混杂系统涉及离散随机过程和连续随机过程之间的耦合。他们在细胞生物学中的应用越来越多，从建模基因网络中的启动子噪声到分析随机门控离子通道对单个神经元和神经网络中电压波动的影响。我们以前推导出的路径积分表示的解决方案，相关的微分查普曼-Kolmogorov方程，狄拉克δ函数的积分表示的基础上，并使用此来确定“最少的行动”的路径中的噪声引起的逃逸从亚稳态。在本文中，我们提出了一种替代推导的路径积分，使用的基础上，括号和"量子力学“运营商。我们展示了如何运营商的方法提供了一个更有效和灵活的框架，用于构建混合路径积分，消除了某些特设的步骤，从以前的推导，并提供了更多的上下文方面的随机路径积分的一般理论。我们还强调了主特征值，光谱间隙和Perron Frobenius定理的重要作用。然后，我们使用摄动方法来开发各种近似计划的混合路径积分和相关的矩生成泛函。首先，我们考虑高斯近似和环路展开的弱噪声限制，类似于量子路径积分的半经典极限。其次，我们通过将随机混合系统视为Ornstein-Uhlenbeck过程的非线性扰动来识别弱耦合极限的模拟。这导致了一个扩展的时刻的产品的自由传播。
英文摘要：
Stochastic hybrid systems involve the coupling between discrete and continuous stochastic processes. They are finding increasing applications in cell biology, ranging from modeling promoter noise in gene networks to analyzing the effects of stochastically-gated ion channels on voltage fluctuations in single neurons and neural networks. We have previously derived a path integral representation of solutions to the associated differential Chapman-Kolmogorov equation, based on integral representations of the Dirac delta function, and used this to determine ``least action'' paths in the noise-induced escape from a metastable state. In this paper we present an alternative derivation of the path integral, based on the use of bra-kets and ``quantum-mechanical'' operators. We show how the operator method provides a more efficient and flexible framework for constructing hybrid path integrals, which eliminates certain ad hoc steps from the previous derivation and provides more context with regards the general theory of stochastic path integrals. We also highlight the important role of principal eigenvalues, spectral gaps and the Perron-Frobenius theorem. We then use perturbation methods to develop various approximation schemes for hybrid path integrals and the associated moment generating functionals. First, we consider Gaussian approximations and loop expansions in the weak noise limit, analogous to the semi-classical limit for quantum path integrals. Second, we identify the analog of a weak-coupling limit by treating the stochastic hybrid system as the nonlinear perturbation of an Ornstein-Uhlenbeck process. This leads to an expansion of the moments in terms of products of free propagators.
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