  Bisimulation is a concept that captures behavioural equivalence of states in
a variety of types of transition systems. It has been widely studied in a
discrete-time setting where the notion of a step is fundamental. In our setting
we are considering "flow"-processes emphasizing that they evolve in continuous
time. In such continuous-time settings, the concepts are not straightforward
adaptations of their discrete-time analogues and we restrict our study to
diffusions that do not lose mass over time and with additional regularity
constraints.
  In previous work we proposed different definitions of behavioural
equivalences for continuous-time stochastic processes where the evolution is a
flow through time. That work only addressed equivalences. In this work, we aim
at quantifying how differently processes behave. We present two pseudometrics
for diffusion-like processes. These pseudometrics are fixpoints of two
different functionals on the space of 1-bounded pseudometrics on the state
space. We also characterize these pseudometrics in terms of real-valued modal
logics; this is a quantitative analogue of the notion of logical
characterization of bisimulation. These real-valued modal logics indicate that
the two pseudometrics are different and thus yield different notions of
behavioural equivalence.
