Abstract

We investigate asymptotic properties of a sequence whose elements are probability distributions which correspond to compositions of independent random operators. We prove the following statements:1) a sequence whose elements are the expected values which corresponds to compositions of independent identically distributed random semigroups converges to a semigroup which is generated by the expected value of the random generator;2) an analog of the law of large numbers for a sequence whose elements are compositions of independent identically distributed random semigroups;3) a sequence whose elements are compositions of random shift operators on position space converges to a semigroup generated by the Laplace operator;4) a sequence whose elements are compositions of random shift operators on position space and random shift operators in momentum space converges to a semigroup generated by the Hamiltonian of the quantum oscillator;5) a sequence whose elements are compositions of random shift operators on phase space converges to a semigroup generated by the Hamiltonian of the quantum oscillator.

中文翻译：

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操作员随机游走和量子振荡器

摘要

我们研究了一个序列的渐近性质，该序列的元素是与独立随机算子的组成相对应的概率分布。我们证明以下陈述：1）一个序列是元素的期望值，该序列对应于独立相等分布的随机半群的组成，收敛到一个由随机发生器的期望值生成的半群; 2）一个模拟定律对于一个序列，其元素由独立的相同分布的随机半群组成; 3）一个序列的元素为位置空间上的随机移位算子组成的序列收敛到由拉普拉斯算子生成的半群;