On the rough-paths approach to non-commutative stochastic calculus

TL;DR

This paper explores applying rough paths theory to non-commutative probability, extending existing results and developing a new integration approach inspired by controlled paths, with implications for free Brownian motion.

Contribution

It extends Lyons' rough paths results to non-commutative settings and introduces an alternative integration method based on controlled paths theory.

Findings

Extended rough paths principles to non-commutative probability

Developed a new non-commutative integration procedure

Derived new approximation results for free Brownian motion

Abstract

We study different possibilities to apply the principles of rough paths theory in a non-commutative probability setting. First, we extend previous results obtained by Capitaine, Donati-Martin and Victoir in Lyons' original formulation of rough paths theory. Then we settle the bases of an alternative non-commutative integration procedure, in the spirit of Gubinelli's controlled paths theory, and which allows us to revisit the constructions of Biane and Speicher in the free Brownian case. New approximation results are also derived from the strategy.