The first-passage area of Wiener process with stochastic resetting

TL;DR

This paper analyzes the statistical properties of the first-passage time and area for a Wiener process with stochastic resetting, providing explicit Laplace transform expressions and moments, advancing understanding of stochastic processes with resets.

Contribution

It introduces explicit formulas for the Laplace transforms and moments of the first-passage time and area for a Wiener process with stochastic resetting, using solutions of associated ODEs.

Findings

Explicit Laplace transform formulas derived

Single and joint moments calculated

Enhanced understanding of stochastic resetting processes

Abstract

For a one-dimensional Wiener process with stochastic resetting ${\cal X}(t)$, obtained from an underlying Wiener process $X(t),$ we study the statistical properties of its first-passage time through zero, when starting from $x>0,$ and its first-passage area, that is the random area enclosed between the time axis and the path of the process ${\cal X} (t)$ up to the first-passage time through zero. By making use of solutions of certain associated ODEs, we are able to find explicit expressions for the Laplace transforms of the first-passage time and the first-passage area, and their single and joint moments.