Duality formula for the bridges of a Brownian diffusion: Application to gradient drifts

In this paper, we consider families of time Markov fields (or reciprocal classes) which have the same bridges as a Brownian diffusion. We characterize each class as the set of solutions of an integration by parts formula on the space of continuous paths C([0;1];Rd). Our techniques provide a characterization of gradient diffusions by a duality formula and, in case of reversibility, a generalization of a result of Kolmogorov.