Modern Stochastics: Theory and Applications

This paper studies Lévy bases on Riemannian manifolds and related processes built from them. We argue that the canonical choice of intensity measure is the Riemannian measure on the manifold, and show how this allows for transporting properties of Lévy bases on ${\mathbb{R}^{d}}$ to the manifold setting via metatimes and subordination constructions. We show how this links to time-changed Lévy processes and how to naturally extend the setting to manifolds with (possibly stochastically) evolving metrics via generalized time-changes. As applications, we introduce so-called M-trawl processes, which generalize the classical trawl processes by letting the trawl operation be induced by a geometric flow, and we discuss the interpretation of Lévy bases on manifolds as Lévy white noise for stochastic partial differential equations on manifolds with Lévy-type forcing.