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

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

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, inPress. ⟨hal-03079171v1⟩