标题： 马尔可夫跳跃过程的路径可观测值的随机微积分：扩散和跳跃动力学的统一 显示英文标题

标题： Stochastic Calculus for Pathwise Observables of Markov-Jump Processes: Unification of Diffusion and Jump Dynamics

摘要： 路径可观测量——随机轨迹的泛函——是时间平均统计力学的核心，并且在热力学不等式如不确定性关系、速度限制和关联界限中起着关键作用。 它们在典型情况下提供了一种热力学推断的方法，当系统中所有耗散自由度无法实验上获取时。 到目前为止，专注于路径可观测量的理论发展主要集中在两个方向：扩散过程和马尔可夫跳跃动力学，几乎是以互不相关的方式进行的。 此外，即使对于扩散和跳跃动力学的相关结果，也是通过一系列不同的方法得出的，这些方法主要是间接的。 最近，随机微积分被证明可以为扩散过程的路径可观测量提供一种直接的方法，而对应的跳跃动力学框架却一直难以捉摸。 在我们的工作中，我们开发了一种与连续空间扩散完全平行的随机微积分，用于马尔可夫跳跃过程的路径可观测量。 我们为跳跃过程提出了一个“朗之万方程”，定义了广义的路径可观测量，并建立了它们的协变结构，从而全面考虑了瞬态和时间非齐次动力学。 我们以最一般的形式证明了已知的热力学不等式，并讨论了饱和条件。 我们确定了路径可观测量对一般（包括热）扰动的响应，并进行了连续极限，以实现扩散和跳跃动力学的完全统一。 我们的结果为离散状态下生成扩散模型的类比以及从波动轨迹中学习随机热力学开辟了新的途径。 显示英文摘要

摘要： Path-wise observables--functionals of stochastic trajectories--are at the heart of time-average statistical mechanics and are central to thermodynamic inequalities such as uncertainty relations, speed limits, and correlation-bounds. They provide a means of thermodynamic inference in the typical situation, when not all dissipative degrees of freedom in a system are experimentally accessible. So far, theories focusing on path-wise observables have been developing in two major directions, diffusion processes and Markov-jump dynamics, in a virtually disjoint manner. Moreover, even the respective results for diffusion and jump dynamics were derived with a patchwork of different approaches that are predominantly indirect. Stochastic calculus was recently shown to provide a direct approach to path-wise observables of diffusion processes, while a corresponding framework for jump dynamics remained elusive. In our work we develop, in an exact parallelism with continuous-space diffusion, a complete stochastic calculus for path-wise observables of Markov-jump processes. We formulate a "Langevin equation" for jump processes, define general path-wise observables, and establish their covariation structure, whereby we fully account for transients and time-inhomogeneous dynamics. We prove the known kinds of thermodynamic inequalities in their most general form and discus saturation conditions. We determine the response of path-wise observables to general (incl. thermal) perturbations and carry out the continuum limit to achieve the complete unification of diffusion and jump dynamics. Our results open new avenues in the direction of discrete-state analogs of generative diffusion models and the learning of stochastic thermodynamics from fluctuating trajectories.