Perturbative expansion for the maximum of fractional Brownian motion.
Perturbative expansion for the maximum of fractional Brownian motion.
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分数布朗运动最大值的微扰展开。
DOI：
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发表时间：
2016
影响因子：
2.4
通讯作者：
KAY Joerg WIESE
中科院分区：
文献类型：
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作者：
Mathieu Delorme;KAY Joerg WIESE
布朗运动是唯一的随机过程，这是高斯，标度不变，和马尔可夫。去掉马尔可夫性质，即，考虑到记忆，我们得到一类称为分数布朗运动的过程，用赫斯特指数H来表示。当H=1/2时，布朗运动被恢复.我们发展了一种微扰方法来处理时间上的非定域性展开在H =H-1/2。这使我们能够导出与极值统计相关的各种观测量的标度指数之外的解析结果：过程的最大值m和达到最大值的时间t_{max}，以及它们的联合分布。我们测试我们的分析预测与广泛的数值模拟不同的H值。它们表现出极好的一致性，即使对于H远离1/2。
Brownian motion is the only random process which is Gaussian, scale invariant, and Markovian. Dropping the Markovian property, i.e., allowing for memory, one obtains a class of processes called fractional Brownian motion, indexed by the Hurst exponent H. For H=1/2, Brownian motion is recovered. We develop a perturbative approach to treat the nonlocality in time in an expansion in ɛ=H-1/2. This allows us to derive analytic results beyond scaling exponents for various observables related to extreme value statistics: the maximum m of the process and the time t_{max} at which this maximum is reached, as well as their joint distribution. We test our analytical predictions with extensive numerical simulations for different values of H. They show excellent agreement, even for H far from 1/2.