We study the extreme value statistics of first-passage trajectories generated from a one-dimensional drifted Brownian motion subject to stochastic resetting to the starting point with a constant rate r. Each stochastic trajectory starts from a positive position x ₀ and terminates whenever the particle hits the origin for the first time. We obtain an exact expression for the marginal distribution Pr(M|x0) of the maximum displacement M. We find that stochastic resetting has a profound impact on Pr(M|x0) and the expected value ⟨M⟩ of M. Depending on the drift velocity v, ⟨M⟩ shows three distinct trends of change with r. For v⩾0 , ⟨M⟩ decreases monotonically with r, and tends to 2x0 as r→∞ . For vc<v<0 , ⟨M⟩ shows a nonmonotonic dependence on r, in which a minimum ⟨M⟩ exists for an intermediate level of r. For v⩽vc , ⟨M⟩ increases monotonically with r. Moreover, by deriving the propagator and using a path decomposition technique, we obtain, in the Laplace domain, the joint distribution of M and the time t_m at which the maximum M is reached. Interestingly, the dependence of the expected value ⟨tm⟩ of t_m on r is either monotonic or nonmonotonic, depending on the value of v. For v>vm , there is a nonzero resetting rate at which ⟨tm⟩ attains its minimum. Otherwise, ⟨tm⟩ increases monotonically with r. We provide an analytical determination of two critical values of v, vc≈−1.69415D/x0 and vm≈−1.66102D/x0 , where D is the diffusion constant. Finally, numerical simulations are performed to support our theoretical results.

中文翻译：

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随机重置下漂移布朗运动第一通道轨迹的极值统计

我们研究一维漂移布朗运动产生的第一通道轨迹的极值统计，该运动以恒定速率随机重置到起点r。每个随机轨迹从正位置开始X ₀并在粒子第一次撞击原点时终止。我们得到了边际分布的精确表达式 磷r（中号|X0） 最大位移中号。我们发现随机重置对 磷r（中号|X0） 和期望值 ⟨中号⟩ 的中号。取决于漂移速度v, ⟨中号⟩ 显示了三种不同的变化趋势r。为了 v⩾0 , ⟨中号⟩ 单调递减r，并且倾向于 2X0 作为 r→无穷大 。为了 vC<v<0 , ⟨中号⟩ 显示出非单调依赖性r，其中最小 ⟨中号⟩ 存在于中等水平r。为了 v⩽vC , ⟨中号⟩ 单调增加r。此外，通过推导传播器并使用路径分解技术，我们在拉普拉斯域中获得了中号和时间Tm_值 此时最大中号到达了。有趣的是，期望值的依赖性 ⟨t米⟩ 的Tm_值 在r是单调的还是非单调的，取决于v。为了 v>v米 ，存在一个非零重置率 ⟨t米⟩ 达到最小值。否则， ⟨t米⟩ 单调增加r。我们提供了两个临界值的分析确定v, vC≈-1.69415D/X0 和 v米≈-1.66102D/X0 ， 在哪里D是扩散常数。最后，进行数值模拟来支持我们的理论结果。