Non-commutative stochastic processes with independent increments

TL;DR

This paper explores the development of non-commutative stochastic processes with independent increments, focusing on quantum probability theory and the influence of Wilhelm von Waldenfels' foundational work.

Contribution

It analyzes how non-commutative notions of independence and Lévy processes evolved from Waldenfels' ideas, emphasizing their mathematical structure.

Findings

Development of non-commutative Lévy processes

Connection between Waldenfels' concepts and quantum probability

Mathematical framework for non-commutative stochastic processes

Abstract

This article is on the research of Wilhelm von Waldenfels in the mathematical field of quantum (or non-commutative) probability theory. Wilhelm von Waldenfels was one of the pioneers, even one of the founders, of quantum probability. We concentrate on a small part of his scientific work. The aspects of physics are practically not mentioned at all. There is nothing on his results in classical probability on groups (Waldenfels operators). This is an attempt to show how the concepts of non-commutative notions of independence and of L\'evy processes on structures like Hopf algebras developed from the ideas of Wilhelm von Waldenfels.