The alternating marked point process of h-slopes of the drifted Brownian motion

TL;DR

This paper characterizes the stationary alternating marked point process formed by the slopes between h-extrema of drifted Brownian motion, extending previous non-drifted results and providing a detailed statistical and local behavior analysis.

Contribution

It extends the understanding of h-extrema slopes to drifted Brownian motion and offers a comprehensive description using Palm--Khinchin theory and renormalization insights.

Findings

Slopes form a stationary alternating marked point process

Complete statistical description of slopes covering the origin

Analysis of Brownian motion behavior near h-extrema

Abstract

We show that the slopes between h-extrema of the drifted 1D Brownian motion form a stationary alternating marked point process, extending the result of J. Neveu and J. Pitman for the non drifted case. Our analysis covers the results on the statistics of h-extrema obtained by P. Le Doussal, C. Monthus and D. Fisher via a Renormalization Group analysis and gives a complete description of the slope between h-extrema covering the origin by means of the Palm--Khinchin theory. Moreover, we analyze the behavior of the Brownian motion near its h-extrema.