Brownian motions, persistent homology and chirality

Abstract Interactions of the maxima and minima of the univariate functions appear in combinatorics as Dyck paths, and in topological data analysis as persistent homology. We study these descriptors for Brownian motions with drift, deriving the intensity measure and correlation functions for the persistence diagram point process $$\textbf{PH}_0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>PH</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:math> , and quantifying the intrinsic asymmetries in the coupling of maxima and minima.