Itô's Formula for Noncommutative $C^2$ Functions of Free Itô Processes with Respect to Circular Brownian Motion

Abstract

In a recent paper, the author introduced a rich class N C k (R) of "noncommutative C k " functions R → C whose operator functional calculus is k-times differentiable and has derivatives expressible in terms of multiple operator integrals (MOIs). In this paper, we explore a connection between free stochastic calculus and the theory of MOIs by proving a "functional free Itō formula" for noncommutative C 2 functions of self-adjoint free Itō processes with respect to circular Brownian motion. To do this, we extend P. Biane and R. Speicher's theory of free stochastic calculus to allow free Itō processes involving circular Brownian motion and its adjoint -instead of only semicircular Brownian motion -and then reinterpret the quantities in the resultant Itō-type formulas as MOIs. As motivation, we also prove a "functional Itō formula" for C 2 functions of Itō processes with respect to n × n matrix Brownian motion.

Brown measure support and the free multiplicative Brownian motion

“…Existence and uniqueness of the solution to (2.6) follows from a standard Picard iteration argument, as in Proposition A.1 of [7]. (The key making the Picard iteration work is the Burkholder-Davis-Gundy estimate for the norm of a free stochastic integral, as in [5,Theorem 3.2.1] or [30,Theorem 3.1.12].) It follows from (2.5) that w s,τ (r) ∼ = w rs,rτ (1), where ∼ = indicates that the two elements have the same * -distribution.…”