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Journal Articles Journal of Applied and Computational Topology Year : 2023

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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Dates and versions

hal-03079171 , version 1 (17-12-2020)
hal-03079171 , version 2 (03-07-2023)

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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⟩
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