On the persistent homology of almost surely $C^0$ stochastic processes
Abstract
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 + ε, ∞[, N x,x+ε and the number of bars of length ≥ ε, N ε. 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 ε and N x,x+ε as ε → 0 and of N ε
as ε → ∞. 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.
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