On the Onsager-Machlup functional for the Brownian motion on the Heisenberg group
Marco Carfagnini, Maria Gordina
TL;DR
This work extends Onsager–Machlup theory to hypoelliptic diffusions on the Heisenberg group by developing a rigorous horizontal-path framework and a sup-norm OM functional. The authors construct continuous horizontal paths via Wong–Zakai-type approximations, define a horizontal projection map $\mathcal{K}$ and a left-invariant semi-metric $d_{\mathcal{H}}$, and apply Cameron–Martin–Girsanov to obtain a precise asymptotic ratio for tube probabilities around a curve $\varphi$. The main contribution is the identification of the domain $D_{\mathcal{L}}=D_{\mathcal{K}}$ and the OM Lagrangian $\mathcal{L}(\varphi)=-\tfrac12 \|\pi_H(\varphi)\|^2_{H_0(\mathbb{R}^2)}$, with a normalization constant $C(\varepsilon)$ governed by the spectral gap of the sub-Laplacian. This provides a pathwise large-deviations description for hypoelliptic diffusions on Carnot groups and offers a framework extendable to other sub-Riemannian settings.
Abstract
Onsager-Machlup functionals are used to describe the dynamics of a continuous stochastic process. For a stochastic process taking values in a Riemannian manifold, they have been studied extensively. We describe the Onsager-Machlup functional with respect to the sup norm for a hypoelliptic Brownian motion on a Heisenberg group. Unlike in the Riemannian case we do not rely on the 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 hypoelliptic Brownian motion, including a new notion of horizontal continuous curves.