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

## TL;DR

This paper develops a path-integral approach to analyze stochastic resetting in Brownian motion, providing exact solutions and demonstrating that energy-dependent resetting enhances confinement efficiency.

## Contribution

It introduces a systematic path-integral method for stochastic resetting with space-dependent rates and applies it to new examples, including energy-dependent resetting in harmonic traps.

## Key findings

- Energy-dependent resetting improves confinement speed.
- Analytical expressions derived for various resetting statistics.
- New insights into stochastic resetting in energy landscapes.

## 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.