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

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.