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 \(]\!-\!\infty ,x]\times [x+\varepsilon ,\infty [\), \(N^{x,x+\varepsilon }\) and the number of bars of length \(\ge \varepsilon \), \(N^\varepsilon \). 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 \(\varepsilon \rightarrow 0\) and of \(N^\varepsilon \) as \(\varepsilon \rightarrow \infty \). 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.
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Acknowledgements
The author would like to thank Pierre Pansu and Claude Viterbo, without whom this work would not have been possible. Many thanks are owed to Jean-François Le Gall and Nicolas Curien, for numerous fruitful discussions relevant to this work as well to the reviewers of this paper for their helpful comments.
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Perez, D. On the persistent homology of almost surely \(C^0\) stochastic processes. J Appl. and Comput. Topology 7, 879–906 (2023). https://doi.org/10.1007/s41468-023-00132-x
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DOI: https://doi.org/10.1007/s41468-023-00132-x
Keywords
- Persistent homology
- Barcodes
- Trees
- Brownian motion
- Stochastic processes
- Markov processes
- Rates of convergence
- Random walks
- Topological data analysis