In 1998 Lyons and Zheng (Asterisque,157-158) observed that the Stratonovich integral of a one form against a reversible and stationary diffusion process makes sense even if the one form is square integrable. Interestingly the resulting process is not a semi-martingale. However, it is a difference of a forward and backward martingale. The technique has lead to a number of useful estimates and is considerably more powerful than an ItF4 approach which would require significant smoothness.

A major failing of the techniques mentioned above, involved the need to start the process with the stationary measure. Integrating the form along the sample path of the process conditioned to start at a pointmight well not make sense. (In 1990 Proc. Royal Soc. Edinburgh Sect. A 115) it was shown by Zheng and Lyons that the integral existed if the one form was bounded. Stoica and Lyons (Annals of Prob 1999) proved that if the one form was in the natural conjectural Lebesgue space (coming from the Sobolev embedding theorem). Results about the behaviour at infinity were also given.

In this talk I will survey the basic results and, if time permits, I will give applications.