What happens when a continuously evolving stochastic process is interrupted with large changes at random intervals $\tau$ distributed as a power law $\sim {\tau }^{-(1+\alpha )};\alpha >0$? Modeling the stochastic process by diffusion and the large changes as abrupt resets to the initial condition, we obtain exact closed-form expressions for both static and dynamic quantities, while accounting for strong correlations implied by a power law. Our results show that the resulting dynamics exhibits a spectrum of rich long-time behavior, from an ever-spreading spatial distribution for $\alpha <1$, to one that is time independent for $\alpha >1$. The dynamics has strong consequences on the time to reach a distant target for the first time; we specifically show that there exists an optimal $\alpha$ that minimizes the mean time to reach the target, thereby offering a step towards a viable strategy to locate targets in a crowded environment.

Stochastic processes offer a fundamentally different paradigm of dynamics than deterministic processes that one is most familiar with, the most prominent example of the latter being Newton’s laws of motion. Here, we discuss in a pedagogical manner a simple and illustrative example of stochastic processes in the form of a particle undergoing standard Brownian diffusion, with the additional feature of the particle resetting repeatedly and at random times to its initial condition. Over the years, many different variants of this simple setting have been studied, including extensions to many-body interacting systems, all of which serve as illustrations of peculiar non-trivial and interesting static and dynamic features that characterize stochastic dynamics at long times. We will provide in this work a brief overview of this active and rapidly evolving field by considering the arguably simplest example of Brownian diffusion in one dimension. Along the way, we will learn about some of the general techniques that a physicist employs to study stochastic processes. Relevant to the special issue, we will discuss in detail how introducing resetting in an otherwise diffusive dynamics provides an explicit optimization of the time to locate a misplaced target through a special choice of the resetting protocol. We also discuss thermodynamics of resetting, and provide a bird’s eye view of some of the recent work in the field of resetting.

We study one-dimensional fluctuating interfaces of length $L$, where the interface stochastically resets to a fixed initial profile at a constant rate $r$. For finite $r$ in the limit $L$\rightarrow${}$\infty${}$, the system settles into a nonequilibrium stationary state with non-Gaussian interface fluctuations, which we characterize analytically for the Kardar-Parisi-Zhang and Edwards-Wilkinson universality class. Our results are corroborated by numerical simulations. We also discuss the generality of our results for a fluctuating interface in a generic universality class.

We study the dynamics of overdamped Brownian particles diffusing in conservative force fields and undergoing stochastic resetting to a given location at a generic space-dependent rate of resetting. We present a systematic approach involving path integrals and elements of renewal theory that allows us to derive analytical expressions for a variety of statistics of the dynamics such as (i) the propagator prior to first reset, (ii) the distribution of the first-reset time, and (iii) the spatial distribution of the particle at long times. We apply our approach to several representative and hitherto unexplored examples of resetting dynamics. A particularly interesting example for which we find analytical expressions for the statistics of resetting is that of a Brownian particle trapped in a harmonic potential with a rate of resetting that depends on the instantaneous energy of the particle. We find that using energy-dependent resetting processes is more effective in achieving spatial confinement of Brownian particles on a faster time scale than performing quenches of parameters of the harmonic potential.