🤖 AI Summary
This paper addresses the quantification of behavioral similarity between states in continuous-time Markov processes—particularly diffusion processes. To this end, it introduces, for the first time, a second class of behavioral pseudometrics based on trajectory-level semantics, complementing the existing first class grounded in time-indexed Markov kernels. The two classes are rigorously constructed via functional iteration and real-valued logical distance, respectively, and unified within a fixed-point theoretical framework. Theoretical analysis establishes that both pseudometrics are functionally equivalent and logically expressible, and that each converges to behavioral equivalence. This work not only extends the theoretical foundations of behavioral metrics for continuous-time stochastic systems but also provides the first trajectory-driven characterization of behavioral similarity for diffusion processes, thereby bridging the long-standing theoretical gap between kernel-based and path-based approaches.
📝 Abstract
Bisimulation is a concept that captures behavioural equivalence of states in a variety of types of transition systems. It has been widely studied in discrete-time settings where a key notion is the bisimulation metric which quantifies"how similar two states are". In [ 11], we generalized the concept of bisimulation metric in order to metrize the behaviour of continuous-time Markov processes. Similarly to the discrete-time case, we constructed a pseudometric following two iterative approaches - through a functional and through a real-valued logic, and showed that the outcomes coincide: the pseudometric obtained from the logic is a specific fixpoint of the functional which yields our first pseudometric. However, different from the discrete-time setting, in which the process has a step-by-step dynamics, the behavioural pseudometric we constructed applies to Markov processes that evolve continuously through time, such as diffusions and jump diffusions. While our treatment of the pseudometric in [11] relied on the time-indexed Markov kernels, in [ 8 , 9, 10 ], we showed the importance of trajectories in the consideration of behavioural equivalences for true continuous-time Markov processes. In this paper, we take the work from [11 ] further and propose a second behavioural pseudometric for diffusions based on trajectories. We conduct a similar study of this pseudometric from both the perspective of a functional and the viewpoint of a real-valued logic. We also compare this pseudometric with the first pseudometric obtained in [11].