Distributional and Extremal Behaviour of Brownian Motion with Exponential Resetting

TL;DR

This paper investigates the distributional and asymptotic characteristics of Brownian motion with drift and exponential resetting, providing explicit formulas and tail behavior analysis for the supremum and infimum of the process.

Contribution

It introduces explicit renewal-type formulas for the supremum distribution and derives asymptotics for the tail distribution of the infimum, advancing understanding of reset Brownian motions.

Findings

Explicit renewal-type formula for supremum distribution

Approximation for the supremum's survival function

Asymptotics of the tail distribution of the infimum

Abstract

We study the distributional and asymptotic properties of the supremum of Brownian motion with drift and exponential resetting. We obtain an explicit renewal-type formula for the distribution of the supremum and then derive an approximation for its survival function. Moreover, we find the asymptotics of the tail distribution of the infimum. We also consider the stationary case and give a new explicit expression for the fidi's of such processes.