Weak Markov Processes as Linear Systems
Abstract
A noncommutative Fornasini-Marchesini system (a multi-variable version of a linear system) can be realized within a weak Markov process (a model for quantum evolution). For a discrete time parameter the resulting structure is worked out systematically and some quantum mechanical interpretations are given. We introduce subprocesses and quotient processes and then the notion of a -extension for processes which leads to a complete classification of all the ways in which processes can be built from subprocesses and quotient processes. We show that within a -extension we have a cascade of noncommutative Fornasini-Marchesini systems. We study observability in this setting and as an application we gain new insights into stationary Markov chains where observability for the system is closely related to asymptotic completeness in a scattering theory for the chain.
Cite
@article{arxiv.1206.0378,
title = {Weak Markov Processes as Linear Systems},
author = {Rolf Gohm},
journal= {arXiv preprint arXiv:1206.0378},
year = {2015}
}
Comments
Expanded version v2 (43 pages) with substantial additions and improvements compared to v1. More details and examples, in particular in sections 3, 4 and 7. Also changes in terminology, compare Def. 3.1, 4.2, 6.4, page 33. To appear in the journal: Mathematics of Control, Signals, and Systems (MCSS)