How to differentiate a quantum stochastic cocycle

TL;DR

This paper reviews two innovative methods for characterizing quantum stochastic cocycles, emphasizing the roles of H"older continuity and holomorphic assumptions to extend traditional differential equation frameworks.

Contribution

It introduces new approaches to infinitesimal characterization of quantum stochastic cocycles, expanding beyond existing quantum stochastic differential equations.

Findings

H"older continuity plays a key role in mapping cocycles on operator spaces.

Holomorphic assumptions enable a broader class of cocycles with infinitesimal characterization.

The approaches extend the scope of quantum stochastic differential equations.

Abstract

Two new approaches to the infinitesimal characterisation of quantum stochastic cocycles are reviewed. The first concerns mapping cocycles on an operator space and demonstrates the role of H\"older continuity; the second concerns contraction operator cocycles on a Hilbert space and shows how holomorphic assumptions yield cocycles enjoying an infinitesimal characterisation which goes beyond the scope of quantum stochastic differential equations.