论文标题
Hermite Trace多项式和混沌分解，用于Hermitian Brownian Motion
Hermite trace polynomials and chaos decompositions for the Hermitian Brownian motion
论文作者
论文摘要
对于非零参数$ Q $，我们定义了HERMITE痕量多项式，这是由排列索引的多元多项式。我们证明了它们的几种组合特性，例如扩展和产品公式。这些跟踪多项式确定的线性函数是$ q = \ frac {1} {n} $的状态，用于$ n $ a a non-Zero Integer。对于这种$ Q $，不同程度的HERMITE痕量多项式是正交的。产品公式扩展到相对于国家的关闭。可以通过$ n \ times n $ hermitian布朗尼动议引起的期望来确定该州。 HERMITE痕量多项式是这种布朗运动的Martingales，而闭合中的元素可以解释为相对于它的随机积分。使用代数上的分级，我们证明了此类积分的几个混乱分解，并分析了相应的创建和歼灭算子。在单变量的，纯痕量多项式情况下，可以用矩阵参数的Hermite多项式来识别痕量的弹性多项式。
For a non-zero parameter $q$, we define Hermite trace polynomials, which are multivariate polynomials indexed by permutations. We prove several combinatorial properties for them, such as expansions and product formulas. The linear functional determined by these trace polynomials is a state for $q = \frac{1}{N}$ for $N$ a non-zero integer. For such $q$, Hermite trace polynomials of different degrees are orthogonal. The product formulas extend to the closure with respect to the state. The state can be identified with the expectation induced by the $N \times N$ Hermitian Brownian motion. Hermite trace polynomials are martingales for this Brownian motion, while the elements in the closure can be interpreted as stochastic integrals with respect to it. Using the grading on the algebra, we prove several chaos decompositions for such integrals, as well as analyze corresponding creation and annihilation operators. In the univariate, pure trace polynomial case, trace Hermite polynomials can be identified with the Hermite polynomials of matrix argument.
