论文标题
ITô的非交换性$ c^2 $ freeitô流程功能的公式
Itô's formula for noncommutative $C^2$ functions of free Itô processes
论文作者
论文摘要
在最近的一篇论文中，作者引入了“非共同$ c^k $”的丰富类$ nc^k（\ mathbb {r}）$ $在本文中，我们通过证明了自偶会自由iTô流程的非交换性$ c^2 $函数的ITô公式，探讨了自由随机演算与MOI理论之间的联系。为此，我们首先扩展了P. biane和R. Speicher的自由随机演算理论（包括其多项式的自由ITô公式），以允许由多维半圆点布朗尼运动驱动的自由iTô进程。然后，在自我伴侣情况下，我们将出现在MOI的多项式中的自由itô公式中出现的对象。这使我们能够扩大一个函数类别的函数类别，并从Biane和Speicher最初考虑的空间（具有两个有限矩的复杂度量的傅立叶变换）到严格较大的空间$ NC^2（\ Mathbb {r}）$。一路上，我们还获得了一个有用的“追踪”ITô公式，用于任意$ c^2 $自由freeitô流程的标量函数。最后，作为动机，我们研究了$ c^2 $标量函数的ITô公式。
In a recent paper, the author introduced a rich class $NC^k(\mathbb{R})$ of "noncommutative $C^k$" functions $\mathbb{R} \to \mathbb{C}$ whose operator functional calculus is $k$-times differentiable and has derivatives expressible in terms of multiple operator integrals (MOIs). In the present paper, we explore a connection between free stochastic calculus and the theory of MOIs by proving an Itô formula for noncommutative $C^2$ functions of self-adjoint free Itô processes. To do this, we first extend P. Biane and R. Speicher's theory of free stochastic calculus -- including their free Itô formula for polynomials -- to allow free Itô processes driven by multidimensional semicircular Brownian motions. Then, in the self-adjoint case, we reinterpret the objects appearing in the free Itô formula for polynomials in terms of MOIs. This allows us to enlarge the class of functions for which one can formulate and prove a free Itô formula from the space originally considered by Biane and Speicher (Fourier transforms of complex measures with two finite moments) to the strictly larger space $NC^2(\mathbb{R})$. Along the way, we also obtain a useful "traced" Itô formula for arbitrary $C^2$ scalar functions of self-adjoint free Itô processes. Finally, as motivation, we study an Itô formula for $C^2$ scalar functions of $N \times N$ Hermitian matrix Itô processes.
