$S^1$-Indexed Brownian Motion Through Abstract Wiener Spaces

Music City Center 

Brownian motion is typically introduced as a stochastic process indexed by the half-closed ray starting at 0, while a Brownian sheet is indexed by an octant of Euclidean space. Recent research has focused on extending these concepts to non-Euclidean index sets (primarily Riemannian manifolds), seeking to define stochastic processes over them that merit the name 'Brownian motion.' This extension is not merely a mathematical exercise - it aims to provide rigorous foundations for the 'SPDE approach' when analyzing data over such spaces, particularly addressing questions about the sparsity of Matérn covariance functions in these settings. In this work, we identify a critical gap in existing approaches: the lack of guaranteed path continuity in the processes explored so far. We present a modification that resolves this limitation, thereby establishing a more robust theoretical foundation for this emerging line of research.

SPDE Approach

Brownian motion

Matérn covariance 

Section on Statistical Computing