On the persistent homology of almost surely C0 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 ]!-!∞,x]× [x+varepsilon,∞[, Nx,x+varepsilon and the number of bars of length ≥ 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 Nx,x+varepsilon as varepsilon → 0 and of N^varepsilon as varepsilon → ∞. 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.