A contractivity approach for probabilistic bisimulations of diffusion processes

This work is concerned with the problem of characterizing and computing probabilistic bisimulations of diffusion processes. A probabilistic bisimulation relation between two such processes is defined through a bisimulation function, which induces an approximation metric on the expectation of the (squared norm of the) distance between the two processes. We introduce sufficient conditions for the existence of a bisimulation function, based on the use of contractivity analysis for probabilistic systems. Furthermore, we show that the notion of stochastic contractivity is related to a probabilistic version of the concept of incremental stability. This relationship leads to a procedure that constructs a discrete approximation of a diffusion process. The procedure is based on the discretization of space and time. Given a diffusion process, we raise sufficient conditions for the existence of such an approximation, and show that it is probabilistically bisimilar to the original process, up to a certain approximation precision.