Quantum stochastic convolution cocycles

TL;DR

This paper introduces and analyzes quantum stochastic convolution cocycles, generalizing quantum Levy processes, and provides characterizations, existence, and uniqueness results within algebraic and operator space frameworks.

Contribution

It develops a comprehensive theory of quantum stochastic convolution cocycles, including their characterization, existence, and dilation, extending classical Levy process concepts to the quantum setting.

Findings

Characterization of stochastic generators for quantum cocycles

Existence and uniqueness of solutions to quantum stochastic differential equations

Dilation results for quantum stochastic convolution cocycles

Abstract

A concept of quantum stochastic convolution cocycle is introduced and studied in two different contexts -- purely algebraic and operator space theoretic. A quantum stochastic convolution cocycle is a quantum stochastic process on a coalgebra satisfying the convolution cocycle relation and the initial condition given by the counit. The notion generalises that of quantum Levy process, which in turn is a noncommutative probability counterpart of classical Levy process on a group. Convolution cocycles arise as solutions of quantum stochastic differential equations. In turn every sufficiently regular cocycle satisfies an equation of that type. This is proved along with the corresponding existence and uniqueness of solutions for coalgebraic quantum stochastic differential equations. The stochastic generators of unital *-homomorphic cocycles are characterised in terms of structure maps on a *-bialgebra. This yields a simple proof of the Schurmann Reconstruction Theorem for a quantum Levy process; it also yields a topological version for a quantum Levy process on a C*-bialgebra. Precise characterisation of the stochastic generators of completely positive and contractive quantum stochastic convolution cocycles in the C*-algebraic context is given, leading to some dilation results. A few examples are presented and some interpretations offered for quantum stochastic convolution cocycles and their stochastic generators on different types of *-bialgebra.