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

Daniel Perez

Published 2020-12-17Version 1

The persistent homology of random processes has been a question of interest in the TDA community. This paper aims to extend a result of Chazal and Divol and to reconcile the probabilistic approach to the study of the topology of superlevel sets of random processes in dimension 1. We provide explicit descriptions of the barcode for Brownian motion and the Brownian bridge. Additionally, we use these results on almost surely $C^0$-processes to describe the barcodes of sequences which admit these objects as limits, up to some quantifiable error.