Stochastic processes on curved spaces

Summary: Stochastic processes are well understood on (flat) Euclidean spaces. But recently there are of interest processes whose state space is on arbitrary (curved) manifold. The notion of semimartingale is well-defined on a manifold and there is actually a succinct connection between those semimartingales on manifolds and semimartingales on Euclidean spaces as long as the manifold has given Riemannian metric or more generally just a connection.

Related problem to above is trying to understand structure of processes on manifolds and seeing how they can be simulated. We will foremost focus on the simplest curved manifold - unit sphere in arbitrary dimension - and on canonical process on it - Brownian motion. It this particular case symmetries of the sphere and additionally invariance of Brownian motion under those symmetries play key role to understanding the process. We wish to utilise this fact to obtain structural consequences for the process. Of particular interest would be so called skew-product decomposition which decomposes a process into two less dimensional and well behaved processes and usually occurs on product spaces where the metric given is not a product one, but so called warped-product one. Using this decomposition one could in theory reduce the process to a series of related one-dimensional processes which are well understood. In particular one can usually simulate one-dimensional processes and turning the decomposition around could yield useful simulation algorithms for the original more dimensional process.

Another problem we wish to tackle is to how to define in a canonical and suitable way a certain class of processes - Levy processes - on general manifolds. Levy processes are originally defined on Euclidean spaces and one particularly interesting feature is that they are essentially simplest class of processes exhibiting jumps. While most of the theory translates to a Lie group setup (Euclidean spaces are in particular also Lie groups), much less is known on what can be done on a general manifold. One of the problems is that a notion of increment does not make sense on a general manifold and addition of jumps is the other problem, since jumps could theoretically take us anywhere on the manifold and manifold are usually well behaved only locally, whereas global structure can be very intricate. There has been some work done on defining some Levy processes on manifolds, but it seems that there are improvements to be made since there seem to exists more processes which would rightfully be dubbed Levy processes and were not included in previous constructions. Our goal will be to use certain tools from differential geometry - principal fibre bundles, connection, frame bundles, (anti-)development - to tackle this problem and try to classify maximal class of processes which could be rightfully called Levy processes on manifolds and this notion should yield classical notion of Levy processes when we consider Euclidean spaces and Lie groups and additionally we should get all possible Levy processes in this setup (which is not the case for current constructions).