论文标题
最佳重置布朗桥
Optimal Resetting Brownian Bridges
论文作者
论文摘要
我们介绍了一个重置的布朗桥，作为一个简单的模型，用于研究搜索时间$ t_f $是有限的，搜索器返回其起点为$ t_f $。这只是一个布朗尼运动，对原点的重置率$ r $ $ r $被限制为在时间上以$ t_f $的起点开始和结束。我们首先提供了一种无拒绝的算法，以在所有维度上生成此类重置桥梁，这是通过带有有效的langevin方程，该方程有效，具有明确的时空依赖性漂移$ \tildeμ（{\ bf x}，t），t）$和重置率$ \ tilde $ \ tilde r（{\ bf x}，\ bf x}，tilde r（\ bf x}，t）$。我们还通过计算精确的各种可观察到的物品（例如均方位移，固定目标的命中概率和预期最大值）来研究搜索过程在一维中的效率。令人惊讶的是，我们发现存在最佳的重置率$ r^*$，即使在存在桥梁约束的情况下，也可以最大化搜索效率。但是，我们表明，桥梁的最佳重置速率负责的物理机制与无桥限制而无需重置布朗尼运动完全不同。
We introduce a resetting Brownian bridge as a simple model to study search processes where the total search time $t_f$ is finite and the searcher returns to its starting point at $t_f$. This is simply a Brownian motion with a Poissonian resetting rate $r$ to the origin which is constrained to start and end at the origin at time $t_f$. We first provide a rejection-free algorithm to generate such resetting bridges in all dimensions by deriving an effective Langevin equation with an explicit space-time dependent drift $\tilde μ({\bf x},t)$ and resetting rate $\tilde r({\bf x}, t)$. We also study the efficiency of the search process in one-dimension by computing exactly various observables such as the mean-square displacement, the hitting probability of a fixed target and the expected maximum. Surprisingly, we find that there exists an optimal resetting rate $r^*$ that maximizes the search efficiency, even in the presence of a bridge constraint. We show however that the physical mechanism responsible for this optimal resetting rate for bridges is entirely different from resetting Brownian motions without the bridge constraint.
