A behavioural pseudometric for continuous-time Markov processes
Linan Chen, Florence Clerc, Prakash Panangaden
TL;DR
This work extends the notion of a bisimulation metric to continuous-time Markov processes with continuous state spaces, addressing the lack of a natural step-wise evolution in continuous time. It develops two equivalent formulations: a fixpoint pseudometric $\overline{\delta}^c$ obtained from a time-discounted transport functional $\mathcal{F}_c$, and a real-valued logic with a formula-based pseudometric $\lambda^c$, proving $\lambda^c = \overline{\delta}^c$. The construction relies on lower semi-continuity, couplings, and Kantorovich duality to compare future distributions $P_t(x)$ and $P_t(y)$ with a discount factor $c\in(0,1)$, and it handles the technical challenges of continuous time via a supremum over $t\ge 0$ and a careful lattice restriction. Through toy and Brownian-motion–based examples, the authors demonstrate the metric’s ability to distinguish behaviors in diffusion processes and discuss practical limitations and potential extensions to trajectory-based analyses.
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.