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

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