UC San Diego
Controlled Rough Paths on Manifolds
- Author(s): Semko, Jeremy
- Advisor(s): Driver, Bruce K
- et al.
We build the foundation for a theory of controlled rough paths
on manifolds. A number of natural candidates for the definition of manifold
valued controlled rough paths are developed and shown to be equivalent. The
theory of controlled rough one-forms along such a controlled path and their
resulting integrals are then defined. This general integration theory does
require the introduction of an additional geometric structure on the manifold
which we refer to as a "parallelism." The
transformation properties of the theory under change of parallelisms is
explored. Using these transformation properties, it is shown that the
integration of a smooth one-form along a manifold valued controlled rough path
is in fact well defined independent of any additional geometric structures. We
present a theory of push-forwards and show how it is compatible with our
integration theory. We give a number of characterizations for solving
a rough differential equation when the solution is interpreted as a controlled
rough path on a manifold and then show such solutions exist and are unique.
We develop the notion of parallel translation along a controlled rough path. This lends itself to a theory of rolling and unrolling maps for not only controlled rough paths but
controlled rough one-forms.