具有自由独立值的非交换广义随机过程Meixner类II . generating function
Meixner class of non-commutative generalized stochastic processes with freely independent values II. The generating function
摘要：
令$T$是其上具有非原子测度$σ$的底层空间。在[{ \it Comm.\数学。物理学} { \bf 292}（2009），99- 129]通过相应正交多项式的连续性刻画了具有自由独立值的非交换广义随机过程Meixner类$ω=(ω(t))_{t\in T}$.在本文中，我们推导出这些正交多项式的生成函数。我们必须回答的第一个问题是：对于无穷多个非对易变量的多项式系统，什么应该作为生成函数？我们构造了一类算子值函数$Z=(Z(t))_{t\in T}$，使得对于任何$s,t\in T$，$Z(t)$与$ω(s)$可交换.则生成函数可以理解为$G(Z,ω)=\sum_{n=0}^\infty \int_{T^n}P^{(n)}(ω(t_1),...,ω(t_n))Z(t_1)...Z(t_n)σ(dt_1)...σ(dt_n)$，其中$P^{(n)}(ω(t_1),...,ω(t_n))$是第$n$次正交多项式（的核）。我们推导出一个明确的形式的$ G(Z,ω)$，它有一个预解式的形式，类似于在经典情况下的生成函数，虽然它涉及到非交换运营商的积分。最后讨论了湮没算符$\partial_t$，$t\in T$作用的一个相关问题.与经典情形相比，我们证明了与自由高斯过程和泊松过程相关的算子$\di_t$具有全局性。这个结果是真正的无限维的，因为在一维中人们失去了整体性的概念。
英文摘要：
Let $T$ be an underlying space with a non-atomic measure $σ$ on it. In [{\it Comm.\ Math.\ Phys.}\ {\bf 292} (2009), 99--129] the Meixner class of non-commutative generalized stochastic processes with freely independent values, $ω=(ω(t))_{t\in T}$, was characterized through the continuity of the corresponding orthogonal polynomials. In this paper, we derive a generating function for these orthogonal polynomials. The first question we have to answer is: What should serve as a generating function for a system of polynomials of infinitely many non-commuting variables? We construct a class of operator-valued functions $Z=(Z(t))_{t\in T}$ such that $Z(t)$ commutes with $ω(s)$ for any $s,t\in T$. Then a generating function can be understood as $G(Z,ω)=\sum_{n=0}^\infty \int_{T^n}P^{(n)}(ω(t_1),...,ω(t_n))Z(t_1)...Z(t_n)σ(dt_1)...σ(dt_n)$, where $P^{(n)}(ω(t_1),...,ω(t_n))$ is (the kernel of the) $n$-th orthogonal polynomial. We derive an explicit form of $ G(Z,ω)$, which has a resolvent form and resembles the generating function in the classical case, albeit it involves integrals of non-commuting operators. We finally discuss a related problem of the action of the annihilation operators $\partial_t$, $t\in T$. In contrast to the classical case, we prove that the operators $\di_t$ related to the free Gaussian and Poisson processes have a property of globality. This result is genuinely infinite-dimensional, since in one dimension one loses the notion of globality.
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