Harmonic coordinates for diffusions in the plane

Abstract

For a class of diffusions X in the plane we construct a global system of coordinates $u(x)$, such that $u(x)$ is close to X at infinity and $u(X)$ is a local martingale. Such coordinates are useful for the study of the long term behaviour of X. The construction uses probabilistic methods, in particular a coupling for general diffusions in the plane.