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

“…The proof of Theorem 1.2 is constructive and it gives an explicit description for a possible choice of a Cartan connection. The result particularly implies that there exist suitable Cartan connections for many interesting examples including all equinilpotentisable contact structures and structures with the growth vector (2,3,5). The latter arise when rolling distributions of surfaces, and it can be applied to modelling rolling of spherical robots on an unknown nonflat ground such as soil.…”

“…Following Métivier [23] 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 [19] and [20]. Already at the level of small-time asymptotics, it is seen that sub-Riemannian diffusions show qualitatively different behaviours compared to Brownian motions.…”

“…This module is not the only possible choice for generating the operator ∆ P , as the following example shows. Let us consider structures with growth vector (2,3,5), that is, the space n is spanned by e 1 , e 2 , e 3 , e 4 , e 5 and we have the following structure equations on g = n ⊕ h The differentials of degree one components are given, for e i ⊗ e j ∧ e k abbreviated to e jk i , by ∂ e 0 ⊗ e 1 = −e 21 1 + e 41 5 − e 51 4 , ∂ e 0 ⊗ e 2 = e 12 2 + e 42 5 − e 52 4 , ∂ e 1 ⊗ e 3 = −e 23 3 − e 12 1 , ∂ e 2 ⊗ e 3 = e 13 3 − e 12 2 , ∂ e 3 ⊗ e 4 = e 14 4 + e 24 5 − e 13 3 , ∂ e 3 ⊗ e 5 = e 15 4 + e 25 5 − e 23 3 .…”