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Journal Article

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


Roldan,  Edgar
Max Planck Institute for the Physics of Complex Systems, Max Planck Society;

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Roldan, E., & Gupta, S. (2017). Path-integral formalism for stochastic resetting: Exactly solved examples and shortcuts to confinement. Physical Review E, 96(2): 022130. doi:10.1103/PhysRevE.96.022130.

Cite as: https://hdl.handle.net/11858/00-001M-0000-002E-1882-2
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.