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].

翻译：互模拟是捕捉各类转移系统中状态行为等价性的概念，已在离散时间设定中得到广泛研究，其中关键概念为量化“两个状态相似程度”的互模拟度量。在[11]中，我们推广了互模拟度量的概念，以度量连续时间马尔可夫过程的行为。类似于离散时间情形，我们通过两种迭代方法——基于泛函和基于实值逻辑——构建了一种伪度量，并证明所得结果一致：从逻辑导出的伪度量是泛函产生的首个伪度量的特定不动点。然而，与具有逐步动态特性的离散时间设定不同，我们所构建的行为伪度量适用于随时间连续演化的马尔可夫过程，如扩散过程和跳跃扩散过程。尽管在[11]中对伪度量的处理依赖于时间索引的马尔可夫核，但在[8,9,10]中，我们证明了轨迹在考虑真实连续时间马尔可夫过程的行为等价性时的重要性。本文进一步拓展[11]的工作，提出基于轨迹的第二种适用于扩散过程的行为伪度量。我们从泛函视角和实值逻辑视角对此伪度量进行了类似研究，并将其与[11]中获得的第一个伪度量进行了比较。