Title: 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 with a generic space-dependent rate of resetting. We present a
systematic approach involving path integrals and elements of renewal theory
that allows 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 timescale than by
performing quenches of parameters of the harmonic potential.