Brownian Manifolds, Negative Type and Geo-temporal Covariances

TL;DR

This paper surveys Brownian manifolds and their relation to geo-temporal covariances, highlighting the role of negative type functions in understanding space-time stochastic processes on manifolds.

Contribution

It provides a comprehensive overview of Brownian manifolds, explores their connection to geo-temporal covariances, and discusses the significance of negative type functions in this context.

Findings

Identification of manifolds that can and cannot parametrize Brownian motion

Analysis of covariances in space-time processes on spherical geometries

Connections established between negative type functions and stochastic processes

Abstract

We survey Brownian manifolds -- manifolds that can parametrise Brownian motion -- and those that cannot. We consider covariances of space-time processes, particularly those when space is the sphere -- geo-temporal processes. There are connections with functions of negative type.