Quantum stochastic convolution cocycles III

TL;DR

This paper extends the theory of quantum Levy processes and stochastic convolution cocycles from compact quantum groups to locally compact quantum groups and multiplier C*-bialgebras, providing a complete characterization of their generators and associated differential equations.

Contribution

It generalizes the framework of quantum stochastic processes to a broader class of quantum groups, establishing existence, uniqueness, and characterization results for their generators.

Findings

Equivalence of Markov-regular quantum Levy processes to quantum stochastic differential equations.

Complete characterization of generating functionals of convolution semigroups on multiplier C*-bialgebras.

Extension of strictness and existence results to locally compact quantum groups.

Abstract

Every Markov-regular quantum Levy process on a multiplier C*-bialgebra is shown to be equivalent to one governed by a quantum stochastic differential equation, and the generating functionals of norm-continuous convolution semigroups on a multiplier C*-bialgebra are then completely characterised. These results are achieved by extending the theory of quantum Levy processes on a compact quantum group, and more generally quantum stochastic convolution cocycles on a C*-bialgebra, to locally compact quantum groups and multiplier C*-bialgebras. Strict extension results obtained by Kustermans, together with automatic strictness properties developed here, are exploited to obtain existence and uniqueness for coalgebraic quantum stochastic differential equations in this setting. Then, working in the universal enveloping von Neumann bialgebra, we characterise the stochastic generators of Markov-regular, *-homomorphic (respectively completely positive and contractive), quantum stochastic convolution cocycles.