In the past few years, stochastic resetting has become a subject of immense interest. Most of the theoretical studies so far focused on instantaneous resetting which is, however, a major impediment to practical realisation or experimental verification in the field. This is because in the real world, taking a particle from one place to another requires finite time and thus a generalization of the existing theory to incorporate non-instantaneous resetting is very much in need. In this paper, we propose a method of resetting which involves non-instantaneous returns facilitated by an external confining trap potential U(x) centered at the resetting location. We consider a Brownian particle that starts its random motion from the origin. Upon resetting, the trap is switched on and the particle starts experiencing a force towards the center of the trap which drives it to return to the origin. The return phase ends when the particle makes a first passage to this center. We develop a general framework to study such a set up. Importantly, we observe that the system reaches a non-equilibrium steady state which we analyze in full details for two choices of U(x), namely, (i) linear and (ii) harmonic. Finally, we perform numerical simulations and find an excellent agreement with the theory. The general formalism developed here can be applied to more realistic return protocols opening up a panorama of possibilities for further theoretical and experimental applications.

在过去的几年中，随机重置已成为人们关注的主题。到目前为止，大多数理论研究都集中在瞬时复位上，然而，瞬时复位是该领域中实际实现或实验验证的主要障碍。这是因为在现实世界中，将粒子从一个地方移到另一个地方需要有限的时间，因此非常需要对现有理论进行概括，以纳入非瞬时重置。在本文中，我们提出了一种重置方法，该方法涉及由外部约束陷阱势能U（x）放在重置位置的中心。我们考虑一个布朗粒子，它从原点开始随机运动。重置后，陷阱被打开，粒子开始受到朝向陷阱中心的力，该力将其驱使返回原点。当粒子第一次到达该中心时，返回阶段结束。我们开发了一个总体框架来研究这种设置。重要的是，我们观察到系统达到了非平衡稳态，我们对U（x），即（i）线性和（ii）谐波。最后，我们进行数值模拟并找到与理论的极好的一致性。此处开发的一般形式主义可以应用于更现实的返回协议，从而为进一步的理论和实验应用开辟了可能性的全景图。