Bisimulation metrics are a successful instrument used to estimate the behavioural distance between probabilistic concurrent systems. They have been defined in both discrete and continuous state space models. However, the weak semantics approach, where non-observable actions are abstracted away, has been adopted only in the discrete case. In this paper we fill this gap and provide a weak bisimulation metric for models with continuous state spaces. A technical difficulty is to provide a suitable notion of weak transition, which requires to lift transitions leaving from states to transitions leaving from a continuous distribution over states. We prove that our weak bisimulation metric is non-expansive, thus allowing for compositional reasoning. We prove that systems at distance zero are equated by a suitable notion of probabilistic weak bisimulation. We apply our theory in a case study where continuous distributions derive from the evolution of the physical environment.

双仿真度量是一种成功的工具，可用于估计概率并发系统之间的行为距离。它们已在离散状态空间模型和连续状态空间模型中定义。但是，仅在离散情况下才采用弱语义方法，其中将不可观察的动作抽象化了。在本文中，我们填补了这个空白，并为具有连续状态空间的模型提供了弱双仿真度量。一个技术难题是提供适当的弱过渡概念，这要求提升从状态离开的过渡到从状态的连续分布离开的过渡。我们证明了我们的弱双仿真度量是不可扩展的，因此可以进行合成推理。我们证明零距离的系统被概率弱双仿真的一个适当的概念等同。我们将我们的理论应用于个案研究中，其中连续分布来自物理环境的演变。