We consider motion of an overdamped Brownian particle subject to stochastic resetting in one dimension. In contrast to the usual setting where the particle is instantaneously reset to a preferred location (say, the origin), here we consider a finite time resetting process facilitated by an external linear potential V(x) = λ|x|(λ > 0). When resetting occurs, the trap is switched on and the particle experiences a force −∂ _x V(x) which helps the particle to return to the resetting location. The trap is switched off as soon as the particle makes a first passage to the origin. Subsequently, the particle resumes its free diffusion motion and the process keeps repeating. In this set-up, the system attains a non-equilibrium steady state. We study the relaxation to this steady state by analytically computing the position distribution of the particle at all time and then analyzing this distribution using the spectral properties of the corresponding Fokker–Planck operator. As seen for the instantaneous resetting problem, we observe a ‘cone spreading’ relaxation with travelling fronts such that there is an inner core region around the resetting point that reaches the steady state, while the region outside the core still grows ballistically with time. In addition to the unusual relaxation phenomena, we compute the large deviation functions (LDFs) associated to the corresponding probability density and find that the LDFs describe a dynamical transition similar to what is seen previously in case of instantaneous resetting. Notably, our method, based on spectral properties, complements the existing renewal formalism and reveals the intricate mathematical structure responsible for such relaxation phenomena. We verify our analytical results against extensive numerical simulations.

我们考虑在一维随机重置的过阻尼布朗粒子的运动。与粒子瞬间重置到首选位置（例如原点）的通常设置相反，这里我们考虑由外部线性势V ( x ) = λ |促进的有限时间重置过程。x |( λ > 0)。当重置发生时，陷阱被打开，粒子受到一个力 -∂ _x V ( x ) 帮助粒子返回到重置位置。一旦粒子第一次到达原点，陷阱就会关闭。随后，粒子恢复其自由扩散运动，该过程不断重复。在此设置中，系统达到非平衡稳态。我们通过分析计算粒子在所有时间的位置分布，然后使用相应 Fokker-Planck 算子的光谱特性分析该分布来研究这种稳态的弛豫。正如瞬时复位问题所见，我们观察到了一个“锥扩展”弛豫，在移动前沿上，在复位点周围有一个内核区域达到稳定状态，而内核外的区域仍然随着时间的推移而弹道增长。除了不寻常的松弛现象外，我们还计算了与相应概率密度相关的大偏差函数 (LDF)，并发现 LDF 描述了类似于之前在瞬时重置情况下看到的动态转变。值得注意的是，我们基于光谱特性的方法补充了现有的更新形式主义，并揭示了导致这种松弛现象的复杂数学结构。我们根据广泛的数值模拟验证了我们的分析结果。补充了现有的更新形式主义，并揭示了导致这种松弛现象的复杂数学结构。我们根据广泛的数值模拟验证了我们的分析结果。补充了现有的更新形式主义，并揭示了导致这种松弛现象的复杂数学结构。我们根据广泛的数值模拟验证了我们的分析结果。