System control and rough paths
by
Terry Lyons
Oxford University, UK

Calculus and its child, the differential equation, are the conventional mathematical tools used to express the evolution of closed systems and relationships between systems (contact transformation and controlled systems).

However, the tools fail when trying to model the response of a system to a rough external control (for example the external influence could be Brownian motion - which is nowhere differentiable).

Ito found a way to mathematize such relationships in the case where the control is Brownian motion but the ``solution" is a measure theoretic object.

Recent developments coming in part from the approach of K.T. Chen have allowed one to define the differential calculus appropriate to control response relationships for a class of objects now known as "rough paths".

We will, with simple examples, explain some of the basic ideas.

Date received: July 9, 2003