标题： 几何布朗运动中对数正态变量的和 显示英文标题

标题： The sum of log-normal variates in geometric Brownian motion

摘要： 几何布朗运动（GBM）是表示自繁殖实体的关键模型。 自繁殖可以被视为生命的定义[5]，它引起的动力学对关注从生物学到经济学的生物系统的人们具有吸引力。 GBM的轨迹按照众所周知的对数正态密度分布，随时间扩展。 然而，在许多应用中，感兴趣的是不是单个轨迹，而是几个轨迹的总和或平均值。 这些对象的分布更为复杂。 在这里，我们展示了两种不同的方法来找到它们的典型轨迹。 我们利用了与自旋玻璃的有趣联系：随机能量模型的期望自由能是对数正态变量的平均值。 我们将这种映射明确化，并发现自由能结果对于GBM轨迹给出了定性正确的行为。 然后我们还使用伊藤微积分计算了对数正态变量的典型总和。 这种方法在定量上与数值工作非常一致。 显示英文摘要

摘要： Geometric Brownian motion (GBM) is a key model for representing self-reproducing entities. Self-reproduction may be considered the definition of life [5], and the dynamics it induces are of interest to those concerned with living systems from biology to economics. Trajectories of GBM are distributed according to the well-known log-normal density, broadening with time. However, in many applications, what's of interest is not a single trajectory but the sum, or average, of several trajectories. The distribution of these objects is more complicated. Here we show two different ways of finding their typical trajectories. We make use of an intriguing connection to spin glasses: the expected free energy of the random energy model is an average of log-normal variates. We make the mapping to GBM explicit and find that the free energy result gives qualitatively correct behavior for GBM trajectories. We then also compute the typical sum of lognormal variates using Ito calculus. This alternative route is in close quantitative agreement with numerical work.