随机重置下的几何布朗运动：一个平稳但非遍历的过程
Geometric Brownian Motion under Stochastic Resetting: A Stationary yet Non-ergodic Process
摘要：
研究了随机重置对几何布朗运动（GBM）的影响。重置是一个进程的突然中断，它连续地更新其动态。我们发现，虽然重置呈现GBM静止，由此产生的过程仍然是非遍历的。令人惊讶的是，重置的效果是关键的非遍历行为的表现。特别是，我们观察到三种不同的长时间制度：淬火状态，不稳定和稳定的退火状态取决于复位强度。值得注意的是，在最后一个机制中，系统是自平均的，因此样本平均值将始终模仿遍历行为，为重置下的GBM建立独立的特征。至关重要的是，上述制度是很好地分离的自平均时间段，可以最小化的最佳复位率。我们的结果可以用来解释数据所产生的股票市场崩溃或重组的投资组合。
英文摘要：
We study the effects of stochastic resetting on geometric Brownian motion (GBM), a canonical stochastic multiplicative process for non-stationary and non-ergodic dynamics. Resetting is a sudden interruption of a process, which consecutively renews its dynamics. We show that, although resetting renders GBM stationary, the resulting process remains non-ergodic. Quite surprisingly, the effect of resetting is pivotal in manifesting the non-ergodic behavior. In particular, we observe three different long-time regimes: a quenched state, an unstable and a stable annealed state depending on the resetting strength. Notably, in the last regime, the system is self-averaging and thus the sample average will always mimic ergodic behavior establishing a stand alone feature for GBM under resetting. Crucially, the above-mentioned regimes are well separated by a self-averaging time period which can be minimized by an optimal resetting rate. Our results can be useful to interpret data emanating from stock market collapse or reconstitution of investment portfolios.
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