Continuous evolution families

TL;DR

This paper explores the properties of continuous evolution families in non-commutative probability, providing equivalence conditions and examples of discontinuous cases to deepen understanding of their structure.

Contribution

It introduces new equivalence conditions for continuous evolution families and presents an example illustrating discontinuity, advancing the theoretical framework.

Findings

Established equivalence conditions for continuous evolution families

Provided an example of a discontinuous evolution family

Enhanced understanding of parameter continuity in non-commutative probability

Abstract

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.