Non-Commutative Stochastic Processes and Bi-Free Probability

TL;DR

This paper explores the relationship between bi-free probability and non-commutative stochastic processes, showing how bi-free tools can model transition operators and derive new formulas for processes with free increments.

Contribution

It introduces a novel application of bi-free probability to model non-commutative stochastic processes and derives new formulas for processes with free increments.

Findings

Transition operators modeled using bi-free probability

Recovered important examples with this approach

Derived new formulas for processes with free increments

Abstract

In this paper, a connection between bi-free probability and the theory of non-commutative stochastic processes is examined. Specifically it is demonstrated that the transition operators for non-commutative stochastic processes can be modelled using technology from bi-free probability. Several important examples are recovered with this approach and new formula are obtained for processes with free increments. The benefits of this approach are also discussed.