On the Onsager-Machlup functional for the Brownian motion on the Heisenberg group

Onsager-Machlup functionals are used to describe the dynamics of a continuous stochastic process. For a Markov process whose generator is an elliptic operator, such functionals have been studied extensively. We describe the Onsager-Machlup functional for a horizontal Brownian motion on a Heisenberg group. This is a Markov process whose generator is a hypoelliptic differential operator. Unlike in the elliptic case we do not rely on tools from differential geometry such as comparison theorems or curvature bounds as these are not easily available in the sub-Riemannian setting. In addition, we study fine properties of trajectories of the horizontal Brownian motion, including a new notion of horizontal continuous curves.