论文标题
带有随机重置的第一通道布朗功能
First-passage Brownian functionals with stochastic resetting
论文作者
论文摘要
我们研究了在随机重置的情况下，一个维度布朗运动的第一学期时间功能的统计特性。首先函数定义为$ v = \ int_0^{t_f} z [x（τ）] $，其中$ t_f $是重置brownian流程$ x（τ）$的第一个 - f $，即第一次交叉零。在这里，粒子以恒定速率$ r $从$ x_0> 0 $开始重置为$ x_r> 0 $，我们专注于以下功能：（i）局部时间$ t_ {loc} = \ int _0^{t_f} d t_f} d τ〜δ τ〜θ（x-x_r）$和（iii）表单$ a_n = \ int _ {0}^{t_f}dτ[x（x（τ）]^n $带有$ n> -2 $的功能。对于前两个功能，我们通过分析得出矩和分布的精确表达式。有趣的是，停留时间时刻以一些最佳的重置速率达到了最小值。在功能性$ a_n $的时刻，也观察到类似的现象。最后，我们表明，大型$ a_n $的$ a_n $分配为$ \ sim \ sim \ sim \ text {exp} \ left（-a_n/a_n/a_n \ right）$ n $的所有值，以及相应的衰减长度$ a_n $也估计。特别是，根据通用观察结果，得出了第一个通过重置（对应于$ n = 0 $案例）的第一个通过时间（对应于$ n = 0 $案例）的确切分布在较大的时间限制下是指数的。由于重置机制减少了不想要的长布朗时首次通道轨迹，因此可以将这种行为漂移从基础过程中进行理解为后果，并导致加速完成。我们通过数值模拟将结果确认为高精度。
We study the statistical properties of first-passage time functionals of a one dimensional Brownian motion in the presence of stochastic resetting. A first-passage functional is defined as $V=\int_0^{t_f} Z[x(τ)]$ where $t_f$ is the first-passage time of a reset Brownian process $x(τ)$, i.e., the first time the process crosses zero. In here, the particle is reset to $x_R>0$ at a constant rate $r$ starting from $x_0>0$ and we focus on the following functionals: (i) local time $T_{loc} = \int _0^{t_f}d τ~ δ(x-x_R)$, (ii) residence time $T_{res} = \int _0^{t_f} d τ~θ(x-x_R)$, and (iii) functionals of the form $A_n = \int _{0}^{t_f} d τ[x(τ)]^n $ with $n >-2$. For first two functionals, we analytically derive the exact expressions for the moments and distributions. Interestingly, the residence time moments reach minima at some optimal resetting rates. A similar phenomena is also observed for the moments of the functional $A_n$. Finally, we show that the distribution of $A_n$ for large $A_n$ decays exponentially as $\sim \text{exp}\left( -A_n/a_n\right)$ for all values of $n$ and the corresponding decay length $a_n$ is also estimated. In particular, exact distribution for the first passage time under resetting (which corresponds to the $n=0$ case) is derived and shown to be exponential at large time limit in accordance with the generic observation. This behavioural drift from the underlying process can be understood as a ramification due to the resetting mechanism which curtails the undesired long Brownian first passage trajectories and leads to an accelerated completion. We confirm our results to high precision by numerical simulations.
