q-Gaussian processes: Non-commutative and classical aspects
q-Gaussian processes: Non-commutative and classical aspects
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DOI：
10.1007/s002200050084
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发表时间：
1997-04-01
影响因子：
2.4
通讯作者：
Speicher, R
中科院分区：
文献类型：
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作者：
Bozejko, M;Kummerer, B;Speicher, R
对于-1<；q<；1，我们研究q-高斯过程，即算子族(非交换随机变量)X-t=a(T)+a(T)*-其中a(T)满足q-对易关系a(S)a(T)*-qa(T)*a(S)=c(S，t)。1对于某些协方差函数c(.，...)-配备真空期望状态。我们证明了在这些过程背后有一个类似于二次量子化的高斯函子的Q-模拟，并且这种结构可以用来将关于Q-高斯过程的问题转化为底层希尔伯特空间中的相应的(而且简单得多的)问题。特别地，我们利用这一思想证明了一大类q-高斯过程具有一种非交换的马尔可夫性，这保证了存在这些非交换过程的经典形式。这回答了Frisch和Bourret[fb]的一个老问题。
We examine, for -1 < q < 1, q-Gaussian processes, i.e. families of operators (non-commutative random variables) X-t = a(t) + a(t)* - where the a(t) fulfill the q-commutation relations a(s)a(t)* - qa(t)*a(s) = c(s, t) . 1 for some covariance function c(.,.) - equipped with the vacuum expectation state. We show that there is a q-analogue of the Gaussian functor of second quantization behind these processes and that this structure can be used to translate questions on q-Gaussian processes into corresponding (and much simpler) questions in the underlying Hilbert space. In particular, we use this idea to show that a large class of q-Gaussian processes possesses a non-commutative kind of Markov property, which ensures that there exist classical versions of these non-commutative processes. This answers an old question of Frisch and Bourret [FB].