Continuous evolution families

Abstract

Recently in relation to the theory of non-commutative probability, the notion of evolution family { ω s , t } s ≤ t \{\omega _{s,t}\}_{s \le t} is generalized assuming only continuity in parameters, namely ( s , t ) ↦ ω s , t (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.