On the persistent homology of almost surely $C^0$ stochastic processes

TL;DR

This paper explores the properties of persistence diagrams derived from almost surely continuous stochastic processes, analyzing their statistical characteristics and asymptotic behaviors, with applications to Brownian motion and related processes.

Contribution

It introduces the analysis of persistence diagram variables for stochastic processes, proving the existence of moment generating functions and studying their asymptotics for semimartingales.

Findings

Variables admit moment generating functions for processes with the strong Markov property.

Asymptotic behavior of diagram variables as epsilon approaches zero and infinity.

Results illustrated with Brownian motion and empirical functions converging to Brownian bridge.

Abstract

This paper investigates the propreties of the persistence diagrams stemming from almost surely continuous random processes on $[0,t]$. We focus our study on two variables which together characterize the barcode : the number of points of the persistence diagram inside a rectangle $]\!-\!\infty,x]\times [x+\varepsilon,\infty[$, $N^{x,x+\varepsilon}$ and the number of bars of length $\geq \varepsilon$, $N^\varepsilon$. For processes with the strong Markov property, we show both of these variables admit a moment generating function and in particular moments of every order. Switching our attention to semimartingales, we show the asymptotic behaviour of $N^\varepsilon$ and $N^{x,x+\varepsilon}$ as $\varepsilon \to 0$ and of $N^\varepsilon$ as $\varepsilon \to \infty$. Finally, we study the repercussions of the classical stability theorem of barcodes and illustrate our results with some examples, most notably Brownian motion and empirical functions converging to the Brownian bridge.