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 $]!-!\infty,x]\times [x+\varepsilon,\infty[$,$N^{x,x+\varepsilon}$andthenumberofbarsoflength$\geq \varepsilon$,$N^\varepsilon$. 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 $N^{\epsilon }$ and $N^{x,x+\epsilon }$ as $\epsilon \to 0$ and of $N^{\epsilon }$ as $\epsilon \to \mathrm{\infty }$. 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.