G-Brownian Motion as Rough Paths and Differential Equations Driven by G-Brownian Motion

The present article is devoted to the study of sample paths of G-Brownian motion and stochastic differential equations (SDEs) driven by G-Brownian motion from the viewpoint of rough path theory. As the starting point, by using techniques in rough path theory, we show that quasi-surely, sample paths of G-Brownian motion can be enhanced to the second level in a canonical way so that they become geometric rough paths of roughness 2 < p < 3. This result enables us to introduce the notion of rough differential equations (RDEs) driven by G-Brownian motion in the pathwise sense under the general framework of rough paths. Next we establish the fundamental relation between SDEs and RDEs driven by G-Brownian motion. As an application, we introduce the notion of SDEs on a differentiable manifold driven by G-Brownian motion and construct solutions from the RDE point of view by using pathwise localization technique. This is the starting point of developing G-Brownian motion on a Riemannian manifold, based on the idea of Eells-Elworthy-Malliavin. The last part of this article is devoted to such construction for a wide and interesting class of G-functions whose invariant group is the orthogonal group. In particular, we establish the generating nonlinear heat equation for such G-Brownian motion on a Riemannian manifold. We also develop the Euler-Maruyama approximation for SDEs driven by G-Brownian motion of independent interest.