概率分布的叠加
Superpositions of Probability Distributions
摘要：
由不同方差v = σ^2的高斯分布叠加得到的概率分布在量子理论和金融市场中起着重要作用。这样的叠加不一定服从马尔可夫过程的Chapman-Kolmogorov半群关系，因为它们可能引入记忆效应。我们导出了v中不破坏半群性质的涂抹分布的一般形式。涂抹技术有两个直接的应用。它允许简化系统的Kramers-Moyal方程的涂抹和未涂抹的条件概率，并可以方便地实现在路径积分演算。在许多情况下，计算路径积分的叠加比计算初始路径积分容易得多。三个简单的例子，并显示如何将该技术扩展到量子力学。
英文摘要：
Probability distributions which can be obtained from superpositions of Gaussian distributions of different variances v = σ^2 play a favored role in quantum theory and financial markets. Such superpositions need not necessarily obey the Chapman-Kolmogorov semigroup relation for Markovian processes because they may introduce memory effects. We derive the general form of the smearing distributions in v which do not destroy the semigroup property. The smearing technique has two immediate applications. It permits simplifying the system of Kramers-Moyal equations for smeared and unsmeared conditional probabilities, and can be conveniently implemented in the path integral calculus. In many cases, the superposition of path integrals can be evaluated much easier than the initial path integral. Three simple examples are presented, and it is shown how the technique is extended to quantum mechanics.
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