Properties of additive functionals of Brownian motion with resetting

TL;DR

This paper analyzes how resetting affects additive functionals of Brownian motion, deriving large deviation principles and explicit distributions for key functionals like occupation time and area.

Contribution

It introduces a variational formula for large deviations with resetting and provides explicit results for specific functionals, advancing understanding of reset Brownian motion.

Findings

Large deviation principle established for additive functionals with resetting

Explicit formulas for distributions and moments of occupation time and area

Resetting significantly alters the distributional properties of functionals

Abstract

We study the distribution of additive functionals of reset Brownian motion, a variation of normal Brownian motion in which the path is interrupted at a given rate and placed back to a given reset position. Our goal is two-fold: (1) For general functionals, we derive a large deviation principle in the presence of resetting and identify the large deviation rate function in terms of a variational formula involving large deviation rate functions without resetting. (2) For three examples of functionals (positive occupation time, area and absolute area), we investigate the effect of resetting by computing distributions and moments, using a formula that links the generating function with resetting to the generating function without resetting.