Path-integral formalism for stochastic resetting: Exactly solved examples and shortcuts to confinement

Abstract

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.

Entropy production in systems with unidirectional transitions

“…Stochastic resetting is a mechanism in which the system undergoes a stochastic dynamics in the state space as well as stochastically resets to a prescribed location with a given transition rate (i.e., a unidirectional process) [16,25,26,[49][50][51][52][53][54][55][56][57][58][59][60][61][62][63][64]. These resetting transitions involve jumps of the system into given locations and can be called the controlled transitions (i.e., these can be tuned from external sources).…”

First passage of a particle in a potential under stochastic resetting: A vanishing transition of optimal resetting rate

“…Recently Stochastic Resetting (SR) in stochastic processes has become a topic of active research [19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37]. In SR problems, a stochastic process is repeatedly returned to its initial position after random time intervals.…”

Integral fluctuation theorems for stochastic resetting systems

“…In contrast, recasting the functional in a form that would involve derivative with respect to x must be done with proper care. This will become important when we try to give a physical interpretation to the functional appearing in (9). This is the subject of the next section.…”

“…Dynamics with resetting, where a system is intermittently returned to a predetermined state, has been fascinating researchers from many fields and disciplines [1][2][3][4][5][6][7][8][9]. Indeed, dynamical systems with resetting have been employed as models for diverse situations such as searching for lost possessions, foraging for food in the wild, stochastic phenotype switching, optimal search algorithms, and random catastrophic events [10][11][12][13][14].…”