Small-time fluctuations for the bridge of a sub-Riemannian diffusion

Abstract

We consider small-time asymptotics for diffusion processes conditioned by their initial and final positions, under the assumption that the diffusivity has a sub-Riemannian structure, not necessarily of constant rank. We show that, if the endpoints are joined by a unique path of minimal energy, and lie outside the sub-Riemannian cut locus, then the fluctuations of the conditioned diffusion from the minimal energy path, suitably rescaled, converge to a Gaussian limit. The Gaussian limit is characterized in terms of the bicharacteristic flow, and also in terms of a second variation of the energy functional at the minimal path, the formulation of which is new in this context.

Cartan connections for stochastic developments on sub-Riemannian manifolds

“…Following Métivier [22] and Ben Arous [7,8], the small-time asymptotics of sub-Riemannian diffusion processes have been studied, amongst others, by Bailleul, Mesnager, Norris [3], by Barilari,Boscain,Neel [4], by de Verdière, Hillairet, Trélat [11] as well as in [18] and [19]. Already at the level of small-time asymptotics, it is seen that sub-Riemannian diffusions show qualitatively different behaviours compared to Brownian motions.…”

“…Moreover, even for equinilpotentisable sub-Riemannian manifolds, there does not always exist a Cartan connection which allows us to characterise the sub-Riemannian diffusion associated to the sub-Laplacian defined with respect to the Popp volume in terms of a stochastic development. However, in Corollary 4.1, we provide a necessary condition to achieve the latter and, in Section 4, we construct suitable Cartan connections for many interesting cases, including three-dimensional contact manifolds, free step two structures, and sub-Riemannian manifolds with growth vector (2,3,5). The last case arises when rolling distributions of surfaces, and it can be applied to modelling rolling of spherical robots on an unknown non-flat ground such as soil.…”

“…We prove Theorem 1.1 and we further show that the Cartan geometry approach recovers the characterisation of Brownian motion on a Riemannian manifold via stochastic development. In Section 4, we explicitly construct Cartan connections on three-dimensional contact manifolds, generic contact structures, free step two structures, and sub-Riemannian manifolds with growth vector (2,3,5) which give rise to the sub-Riemannian diffusion processes associated with the sub-Laplacians defined with respect to the Popp volume.…”

“…Sub-Riemannian manifolds with growth vector(2,3,5) For the final example, which results in the proof of Proposition 4.5, we consider sub-Riemannian manifolds with growth vector (2, 3, 5), which arise as rolling distributions of surfaces.The nilpotent Lie algebra n is the span of the elements e 1 , e2 , e 3 , e 4 , e 5 satisfying the following structure equations [e 1 , e 2 ] = e 3 , [e 1 , e 3 ] = e 4 , [e 2 , e 3 ] = e 5 .…”

On hypoelliptic bridge

“…On the other hand the following results are proved recently: the Brownian bridge concentrates on the sub-Riemannian geodesic at t → 0. See I. Bailleul, L. Mesnager and J. Norris [2], and Y. Inahama [31]. Since the semi-martingale property depends on properties of the heat kernel for small time, and since the sub-Riemannian geodesic is horizontal in whose direction the singularity in t should be exactly t − n 2 , we tend to believe this semi-martingale property holds much more generally.…”