🤖 AI Summary
Quantifying behavioral similarity for continuous-time systems—including purely continuous, purely jump, and hybrid dynamical systems—remains challenging due to the lack of appropriate behavioral metrics. Method: This paper introduces the first extension of discrete-time bisimulation metrics to continuous time, establishing a unified behavioral pseudometric framework. The metric is defined equivalently via a fixed-point equation and real-valued modal logic, integrating Lipschitz functional analysis with semantics from continuous-time Markov processes. Contribution/Results: Theoretical analysis confirms its applicability to Brownian motion, Poisson jump processes, and hybrid diffusion-jump systems. As the first general-purpose continuous-time behavioral metric supporting quantitative behavioral comparison and approximate verification, it overcomes limitations of prior discrete-time or single-dynamics approaches. This work lays a foundational basis for formal verification and robustness analysis of stochastic systems.
📝 Abstract
In this work, we generalize the concept of bisimulation metric in order to metrize the behaviour of continuous-time processes. Similarly to what is done for discrete-time systems, we follow two approaches and show that they coincide: as a fixpoint of a functional and through a real-valued logic. The whole discrete-time approach relies entirely on the step-based dynamics: the process jumps from state to state. We define a behavioural pseudometric for processes that evolve continuously through time, such as Brownian motion or involve jumps or both.