Transience of diffusions on Heisenberg and Grushin distributions

Abstract

In the absence of a general theory of diffusions on non-integrable distributions, an important role is played by the investigation of some particular examples. This paper deals with a couple of these type of examples. The first one is the Heisenberg diffusion, which is a degenerate diffusion with non-holonomic constraints living on the horizontal distribution of the Heisenberg group. The paper proves the transience property of the Heisenberg diffusion for small balls. The second example is the Grushin diffusion, also a degenerate diffusion, which moves in the plane along the Grushin distribution. We prove the transience property of this diffusion.