含时空力反常过程的Feynman-Kac方程
Feynman-Kac equation for anomalous processes with space- and time-dependent forces
摘要：
随机过程Y（t）的泛函模拟了许多物理上的时间扩展的可观测量，例如粒子位置、局部和占据时间或累积的机械功。当Y（t）为正态扩散过程时，它们的统计量可作为Feynman-Kac方程的解得到。这个方程提供了扩散过程的期望值和确定性二阶偏微分方程的解之间的关键联系。当Y（t）是一个反常扩散过程时，最近已经导出了费曼-卡茨方程的推广，它包含了幂律或更一般的潜在随机游走的等待时间分布。这种等待时间的一般表示是用Lévy过程来表示的，该过程的拉普拉斯指数与广义Feynman-Kac方程中出现的记忆核相关。相应的异常过程已被证明可以捕获非线性均方位移，表现出不同标度制度之间的交叉，这已经在生物系统中观察到，如细胞内环境中的迁移细胞或扩散大分子。然而，空间和时间相关的力量驱动的广义反常过程的动力学的情况下还没有得到解决。在这里，我们提出了缺失的Feynman-Kac方程在这种情况下，通过使用从属技术的推导。此外，我们讨论了它的扩展显着依赖于时间的泛函，这是相关的随机热力学的反常扩散系统。得到了一个简单非平衡模型的功涨落的精确结果。在本文中，我们还提供了一个教学介绍Lévy过程，半鞅及其相关的随机微积分，这背后的数学公式的异常扩散作为一个从属过程。
英文摘要：
Functionals of a stochastic process Y(t) model many physical time-extensive observables, e.g. particle positions, local and occupation times or accumulated mechanical work. When Y(t) is a normal diffusive process, their statistics are obtained as the solution of the Feynman-Kac equation. This equation provides the crucial link between the expected values of diffusion processes and the solutions of deterministic second-order partial differential equations. When Y(t) is an anomalous diffusive process, generalizations of the Feynman-Kac equation that incorporate power-law or more general waiting time distributions of the underlying random walk have recently been derived. A general representation of such waiting times is provided in terms of a Lévy process whose Laplace exponent is related to the memory kernel appearing in the generalized Feynman-Kac equation. The corresponding anomalous processes have been shown to capture nonlinear mean square displacements exhibiting crossovers between different scaling regimes, which have been observed in biological systems like migrating cells or diffusing macromolecules in intracellular environments. However, the case where both space- and time-dependent forces drive the dynamics of the generalized anomalous process has not been solved yet. Here, we present the missing derivation of the Feynman-Kac equation in such general case by using the subordination technique. Furthermore, we discuss its extension to functionals explicitly depending on time, which are relevant for the stochastic thermodynamics of anomalous diffusive systems. Exact results on the work fluctuations of a simple non-equilibrium model are obtained. In this paper we also provide a pedagogical introduction to Lévy processes, semimartingales and their associated stochastic calculus, which underlie the mathematical formulation of anomalous diffusion as a subordinated process.
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