论文标题
随机重置的非均匀随机步行：吉利斯模型的应用
Non-homogeneous random walks with stochastic resetting: an application to the Gillis model
论文作者
论文摘要
我们考虑在存在重置的情况下，第一个通过空间非均匀随机行走的起源的问题，其依赖位置的漂移，称为吉利斯随机行走。步行从初始站点开始$ x_0 $，并且固定概率$ r $，每个步骤都可以将其移至给定的站点$ x_r $。从一般的角度来看，我们首先获得了有关第一个打击时间分布的第一刻和第二刻的一系列结果，对广泛的过程有效，包括缺乏翻译不变性属性的随机步行；然后，我们将这些结果应用于特定模型。当不应用重置时，通过调整定义该过程的过渡概率的参数的值，用$ε$表示，步行的复发属性会更改，我们可以观察到：瞬态步行，无效的步行，或正面的步行。当打开重置机制时，我们在所有制度中进行定量研究搜索效率的提高。特别是，在每种情况下，重置都允许系统以概率为1，并且平均在有限的时间内到达目标。如果无重置系统处于瞬态或无效的状态，这使得重置始终有利且此外，它可以确保存在最佳的重置概率$ r^* $，从而最大程度地减少了第一次打击时间。取而代之的是，当系统是积极的，重置的引入不一定是有益的。我们解释说，在这种情况下，存在一个阈值$ r _ {\ mathrm {th}} $对于重置概率$ r $，高于该概率的重置机制在此概率上产生了更大的平均平均时间，相对于无重置系统。我们提供了$ r _ {\ mathrm {th}} $的研究，对于系统参数的某些值可能为零，这意味着...
We consider the problem of the first passage time to the origin of a spatially non-homogeneous random walk with a position-dependent drift, known as the Gillis random walk, in the presence of resetting. The walk starts from an initial site $ x_0 $ and, with fixed probability $ r $, at each step may be relocated to a given site $ x_r $. From a general perspective, we first derive a series of results regarding the first and the second moment of the first hitting time distribution, valid for a wide class of processes, including random walks lacking the property of translational invariance; we then apply these results to the specific model. When resetting is not applied, by tuning the value of a parameter which defines the transition probability of the process, denoted by $ ε$, the recurrence properties of the walk are changed, and we can observe: a transient walk, a null-recurrent walk, or a positive-recurrent walk. When the resetting mechanism is switched on, we study quantitatively in all regimes the improvement of the search efficiency. In particular, in every case resetting allows the system to reach the target with probability one and, on average, in a finite time. If the reset-free system is in the transient or null-recurrent regime, this makes resetting always advantageous and moreover, it assures the existence of an optimal resetting probability $ r^* $ which minimizes the mean first hitting time. Instead, when the system is positive-recurrent, the introduction of resetting is not necessarily beneficial. We explain that in this case there exists a threshold $ r_{\mathrm{th}} $ for the resetting probability $ r $, above which the resetting mechanism yields a larger mean first hitting time with respect to the reset-free system. We provide a study of $ r_{\mathrm{th}} $, which can be zero for some values of the system parameters, meaning that...
