  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.
