From minimal embeddings to minimal diffusions

We show that there is a one-to-one correspondence between diffusions and the solutions of the Skorokhod Embedding Problem due to Bertoin and Le-Jan. In particular, the minimal embedding corresponds to a ‘minimal local martingale diffusion’, which is a notion we introduce in this article. Minimality is closely related to the martingale property. A diffusion is minimal if it it minimises the expected local time at every point among all diffusions with a given distribution at an exponential time. Our approach makes explicit the connection between the boundary behaviour, the martingale property and the local-time characteristics of time-homogeneous diffusions.