Bisimulation for Feller-Dynkin Processes

Abstract

Bisimulation is a concept that captures behavioural equivalence. It has been studied extensively on nonprobabilistic systems and on discrete-time Markov processes and on so-called continuous-time Markov chains. In the latter time is continuous but the evolution still proceeds in jumps. We propose two definitions of bisimulation on continuoustime stochastic processes where the evolution is a flow through time. We show that they are equivalent and we show that when restricted to discrete-time, our concept of bisimulation encompasses the standard discrete-time concept. The concept we introduce is not a straightforward generalization of discrete-time concepts.

Behavioural equivalences for continuous-time Markov processes

“…Depending on which one is adapted, we get either temporal equivalence or bisimulation. In Chen et al (2019), we introduced only the notion of "bisimulation" where the induction condition is adapted to trajectories.…”

“…The core idea is that an equivalence can be viewed as a span or cospan of morphisms. This work was published in Chen et al (2019).…”

Towards a Classification of Behavioural Equivalences in Continuous-time Markov Processes

“…In [6,7,8], we proposed different notions of behavioural equivalences on continuous-time processes. We showed that there were several possible extensions of the notion of bisimulation to continuous time and that the continuous-time notions needed to involve trajectories in order to be meaningful.…”

“…In our setting we are considering "flow"-processes emphasizing that they evolve in continuous time. In such continuous-time settings, the concepts are not straightforward adaptations of their discrete-time analogues and we restrict our study to diffusions that do not lose mass over time and with additional regularity constraints.In [6,7] we proposed different definitions of behavioural equivalences for continuous-time stochastic processes where the evolution is a flow through time. That work only addressed equivalences.…”

“…In [6,7] we proposed different definitions of behavioural equivalences for continuous-time stochastic processes where the evolution is a flow through time. That work only addressed equivalences.…”