Two behavioural pseudometrics for continuous-time Markov processes
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arXiv
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| Natura: | Preprint |
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2025
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| _version_ | 1866917106685050880 |
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| author | Chen, Linan Clerc, Florence Panangaden, Prakash |
| author_facet | Chen, Linan Clerc, Florence Panangaden, Prakash |
| contents | 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]. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_21621 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Two behavioural pseudometrics for continuous-time Markov processes Chen, Linan Clerc, Florence Panangaden, Prakash Logic in Computer Science 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]. |
| title | Two behavioural pseudometrics for continuous-time Markov processes |
| topic | Logic in Computer Science |
| url | https://arxiv.org/abs/2511.21621 |