Time reversal of reflected Brownian motion with Poissonian resetting

TL;DR

This paper investigates the time reversal of reflecting Brownian motion with Poissonian resetting, revealing that the reversed process is a Brownian motion with negative drift and non-local boundary conditions, with implications for boundary behavior and elliptic problems.

Contribution

It provides a novel probabilistic description of the time-reversed process, characterizes its dynamics, and explores boundary interactions in the context of Poissonian resetting.

Findings

Reversed process is a Brownian motion with negative drift.

Both original and reversed processes leave the same boundary traces.

The study links the process dynamics to elliptic boundary value problems.

Abstract

In this paper, we study reflecting Brownian motion with Poissonian resetting. After providing a probabilistic description of the phenomenon using jump diffusions and semigroups, we analyze the time-reversed process starting from the stationary measure. We prove that the time-reversed process is a Brownian motion with a negative drift and non-local boundary conditions at zero. Moreover, we further study the time-reversed process between two consecutive resetting points and show that, within this time window, it behaves as the same reflecting Brownian motion with a negative drift, where both the jump sizes and the time spent at zero coincide with those of the process obtained under the stationary measure. We characterize the dynamics of both processes, their local times, and finally investigate elliptic problems on positive half-spaces, showing that the two processes leave the same traces at the boundary.