离散马尔可夫演化的一种泛伸缩
A universal dilation of discrete Markov evolutions
摘要：
给定一个有限状态空间E，我们为E上所有可能的离散时间马尔可夫链（齐次或非齐次）建立一个泛伸缩：我们引入了第二个系统（“环境”）和系统E在此环境下的确定性可逆时间齐次全局演化，使得E的任何马尔可夫演化都可以通过适当选择环境的初始（随机）状态来实现，其因此确定系统的转移概率。我们还比较了这种膨胀与量子动力学半群的量子膨胀：给定一个经典马尔可夫半群，我们表明，它可以扩展到一个量子动力学半群，我们可以找到一个量子膨胀到一组 *-自同构承认一个不变的阿贝尔子代数，这个量子膨胀只是我们的经典膨胀。
英文摘要：
Given a finite state space E, we build a universal dilation for all possible discrete time Markov chains on E, homogeneous or not: we introduce a second system (an ``environment'') and a deterministic invertible time-homogeneous global evolution of the system E with this environment such that any Markov evolution of E can be realized by a proper choice of the initial (random) state of the environment, which therefore determines the transition probabilities of the system. We also compare this dilation with the quantum dilations of a Quantum Dynamical Semigroup: given a Classical Markov Semigroup, we show that it can be extended to a Quantum Dynamical Semigroup for which we can find a quantum dilation to a group of *-automorphisms admitting an invariant abelian subalgebra where this quantum dilation gives just our classical dilation.
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