存在测量误差和噪声时随机游动的破纪录统计量
Record-breaking statistics for random walks in the presence of measurement error and noise
摘要：
我们解决的问题的距离记录设置的随机步行者在测量误差，$δ$，和加性噪声，$γ$的存在下，并显示（上）记录的平均数$n$步骤仍然普遍增长为$< R_n> \sim n^{1/2}$为大$n$的所有跳跃分布，包括Lévy航班，并为所有$δ$和$γ$。与1/2的普遍增长指数相反，由$n^{1/2}$的前因子衡量的创纪录速度取决于$δ$和$γ$。在没有噪声（$γ=0$）的情况下，前因子$S(δ)$针对任意跳跃分布被明确地评估，并且其随着$δ$的增加而单调地减小，而在完美测量$(δ=0)$的情况下，对应的前因子$T(γ)$随着$γ$而增加。我们的分析结果支持广泛的数值模拟和定性相似的结果被发现在二维和三维。
英文摘要：
We address the question of distance record-setting by a random walker in the presence of measurement error, $δ$, and additive noise, $γ$ and show that the mean number of (upper) records up to $n$ steps still grows universally as $< R_n> \sim n^{1/2}$ for large $n$ for all jump distributions, including Lévy flights, and for all $δ$ and $γ$. In contrast to the universal growth exponent of 1/2, the pace of record setting, measured by the pre-factor of $n^{1/2}$, depends on $δ$ and $γ$. In the absence of noise ($γ=0$), the pre-factor $S(δ)$ is evaluated explicitly for arbitrary jump distributions and it decreases monotonically with increasing $δ$ whereas, in case of perfect measurement $(δ=0)$, the corresponding pre-factor $T(γ)$ increases with $γ$. Our analytical results are supported by extensive numerical simulations and qualitatively similar results are found in two and three dimensions.
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