Continuous evolution families

Recently in relation to the theory of non-commutative probability, a notion of evolution families $\{{\omega }_{s,t}{\}}_{s\le t}$ is generalized that are only continuous in parameters, namely $(s,t)\mapsto {\omega }_{s,t}$ is continuous with respect to locally uniform convergence on a planar domain. In this article we present various equivalence conditions to the continuous evolution families concerned with the left and right parameters. We also provide an example of a discontinuous evolution family in the last section.