Framework for fluctuating times and counting observables in stochastic excursions

Many natural systems exhibit dynamics characterized by alternating phases or recurring sets of states. Describing the fluctuations of such systems over stochastic trajectories is necessary across diverse fields, from biological motors to quantum thermal machines. In an accompanying letter, we introduced the notion of stochastic excursions-a framework to analyze out of equilibrium processes via subtrajectories. Through counting observables, this framework captures finite-time fluctuations and trajectory-level behavior, which provides insights into thermodynamical tradeoffs between thermodynamic quantities of interest, such as entropy production and dynamical activity. In this work, we enhance this formalism by providing a suite of technical results on how to efficiently compute excursion-related quantities. Our analytical results provide explicit formulas for general moments of counting variables and excursion duration, as well as their covariance and conditional moments. We show that excursion counting statistics recover full counting statistics results, and uncover a relation between fluctuations of counting observables at the single-excursion level and the steady-state diffusion coefficient (noise). We also discuss a fluctuation theorem for individual excursions and show that it implies a modified thermodynamic uncertainty relation at the excursion level. In addition, we explore how analyzing excursions and using the results developed here can yield insights into three problems of interest: the three-qubit absorption refrigerator, cellular sensing, and birth-and-death processes.