Temporal-Logical Stochastic Processes

Abstract

We develop a measure-theoretic framework for Temporal-Logical Stochastic Processes (TLSP), in which temporal-logic specifications induce time-indexed stochastic observables capturing the robustness of temporal satisfaction. Building upon existing probabilistic interpretations of temporal-logic robustness, we systematically investigate the stochastic-process structure of these observables. We establish their measurability, shifted-filtration adaptedness, and sample-path regularity under suitable assumptions, and characterize the action of temporal operators on stochastic processes. We derive pathwise perturbation bounds and quantify the sensitivity of temporal satisfaction probabilities through the probability mass concentrated near the logical boundary. A topology on the observable-process space and a semantic pseudometric on temporal formulas are introduced to establish convergence of temporal-logical observables under perturbations of formula parameters and temporal intervals. We further characterize conditions ensuring weak convergence of observable-process distributions and convergence of temporal satisfaction probabilities. Finally, we establish a process-level large-deviation principle for temporal-logical observables, providing a variational characterization of rare temporal events through minimum-action trajectories. The resulting framework connects temporal-logic semantics with stochastic-process topology, stability, convergence, and rare-event asymptotics, providing a mathematical foundation for the analysis of temporally constrained stochastic systems.