时间积分泛函的 Feynman-Kac 理论：Itô 微积分与泛函微积分的对比
Feynman-Kac theory of time-integrated functionals: Itô versus functional calculus
AI总结：
本文直接通过 Itô 微积分和泛函微积分推导了一般加性动力学泛函的 Feynman-Kac 理论，并利用 Dyson 级数重新概括了稳态方差结果，旨在统一该领域不同方法并提供更易理解的路径。
摘要：
动力学泛函（如经验密度和流，以及随机热力学中的热量、功和广义流）的涨落通常在 Feynman-Kac 倾斜形式体系内进行研究。该形式体系在物理学文献中通常通过某种形式的 Kramers-Moyal 展开来推导，或在数学文献中通过 Cameron-Martin-Girsanov 方法来推导。在此，我们直接通过 Itô 微积分和泛函微积分推导了一般加性动力学泛函的 Feynman-Kac 理论，其中后者的结果实际上似乎是全新的。随后，我们利用 Dyson 级数独立地重新概括了 Dieball 和 Godec（2022 Phys. Rev. Lett. 129 140601）以及 Dieball 和 Godec（2022 Phys. Rev. Res. 4 033243）中最近推导的关于一般加性动力学泛函的稳态（协）方差的结果。我们希望我们的工作能将该领域中所采用的动力学泛函统计的不同方法置于一个共同的基础之上，并阐明更易于理解的倾斜形式体系路径。
英文摘要：
The fluctuations of dynamical functionals such as the empirical density and current as well as heat, work and generalized currents in stochastic thermodynamics are usually studied within the Feynman-Kac tilting formalism, which in the Physics literature is typically derived by some form of Kramers-Moyal expansion, or in the Mathematical literature via the Cameron-Martin-Girsanov approach. Here we derive the Feynman-Kac theory for general additive dynamical functionals directly via Itô calculus and via functional calculus, where the latter result in fact appears to be new. Using Dyson series we then independently recapitulate recent results on steady-state (co)variances of general additive dynamical functionals derived recently in Dieball and Godec ({2022 \textit{Phys. Rev. Lett.}~\textbf{129} 140601}) and Dieball and Godec ({2022 \textit{Phys. Rev. Res.}~\textbf{4} 033243}). We hope for our work to put the different approaches to the statistics of dynamical functionals employed in the field on a common footing, and to illustrate more easily accessible ways to the tilting formalism.
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