MSc Defence by Mathias Højgaard Jensen
Stochastic Processes in Manifolds and Fiber Bundles
Continuous semimartingales are the most natural stochastic processes within stochastic calculus. They arise as solutions to stochastic differential equations and they extend easily into a differential geometric setting. Pathwise existence and uniqueness results of solutions to stochastic differential equations in Euclidean space are examined. By means of embedding it is possible to obtain existence and uniqueness results for manifold-valued stochastic differ- ential equations driven by Euclidean-valued semimartingales. This extrinsic approach is shown to coincide with the intrinsic approach yielding a method of obtaining continuous semimartingales on a manifold from Euclidean ones. Endowing the manifold with a connection establishes a natural relation with the horizontal part of the frame bundle. This lends itself to the definition of horizontal semimartingales by generalizing the horizontal lift of smooth curves on manifolds.
Following this, a one-to-one correspondence between Euclidean semimartingales and manifold-valued ones are obtained through horizontal lift. Furthermore, this one-to-one correspondence allows for a notion of manifold valued martingales to be defined. Their relation to diffusion processes are examined and an equivalent definition is obtained by means of diffusion theory. By exploiting the extra structure given by a Riemannian manifold, a manifold valued Brownian motion is defined as a diffusion process with diffusion operator being a manifold valued generalization of the usual Laplace operator. Thus, it is shown that a manifold-valued Brownian motion is a manifold-valued martingale and a one-to-one correspondence with its Euclidean counterpart is established.
This also provides a definition of horizontal Brownian motion on the orthonormal frame bundle. In conclusion, stochastic processes extends naturally to the context of differential geometry.