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

Daniel Perez (1, 2, 3)

1 DATASHAPE - Understanding the Shape of Data
2 LMO - Laboratoire de Mathématiques d'Orsay
3 DMA - Département de Mathématiques et Applications - ENS Paris

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.

Domaines: Probabilités [math.PR]; Géométrie algorithmique [cs.CG]; Topologie algébrique [math.AT]

Daniel Perez. On the persistent homology of almost surely $C^0$ stochastic processes. Journal of Applied and Computational Topology, 2023, ⟨10.1007/s41468-023-00132-x⟩. ⟨hal-03079171v2⟩