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

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.