Large deviations for small noise hypoelliptic diffusion bridges on sub-Riemannian manifolds
Hypoelliptic diffusion processes on a compact sub-Riemannian manifold \((\mathcal{M}, \mathcal{D}, g)\) of the form \N\[\NdX^\varepsilon_t = \varepsilon \sum_{i=1}^{d} V_i(X^\varepsilon_t) \circ dw^i_t + \varepsilon^2 V_0(X^\varepsilon_t) dt, \quad X^\varepsilon_0 = x, \N\]\Nwith \(\varepsilon \to 0\), are studied under the (strong) Hörmander type conditions for ``smooth enough'' vector fields \((V_i, 0\le i\le d)\). Here \(d\) is the dimension of the fiber bundle \(\mathcal{D} = (\mathcal{D}_x, x\in \mathcal{M})\), where the latter defines the sub-Riemannian structure togethter with the inner product \(g=(g_x, x\in \mathcal{M})\). Assuming that the process has an everywhere positive density, the large deviation (LD) principle is established for the bridge on \(0\le t\le 1\) conditioned by \(X^\varepsilon_1=a\), as \(\varepsilon \downarrow 0\). The rate function, or the action functional in the space \(C([0,1];\mathcal{M})\), is proved to have the form \N\[\NJ(\gamma) = \frac12(\mathcal{E}(\gamma) - d_{SR}(x,a)^2), \N\]\Nwhere \(\mathcal{E}(\gamma)\) is the ``energy'' of the continuous (differentiable) path \(\gamma\) on \(\mathcal{M}\) with \(\gamma_0=x, \gamma_1=a\), associated with the sub-Riemannian metric, and \(d_{SR}(x,a)\) stands for the corresponding distance between the two points in \(\mathcal{M}\). The positivity of the density is guaranteed by the lemma in the appendix.\N\NThe following main tools are used, among other references: Freidlin-Wentzell's LD theory, Watanabe's generalized Wiener functionals, Eells-Elworthy's stochastic parallel transport construction, ``manifold-valued'' Malliavin's calculus by Taniguchi, and the rough paths theory. \N\NThe paper generalizes earlier results on large deviations for diffusion bridges: [\textit{P. Hsu}, Probab. Theory Relat. Fields 84, No. 1, 103--118 (1990; Zbl 0698.58052)] for pinned Brownian motion, [Trans. Am. Math. Soc. 367, No. 11, 8107--8137 (2015; Zbl 1338.60083)] (by the author) for pinned diffusions in Euclidean spaces, [\textit{I. Bailleul}, Lect. Notes Math. 2168, 189--198 (2016; Zbl 1372.60027)] for pinned diffusions on sub-Riemannian manifolds based on a different analytic approach.
scientific article; zbMATH DE number 7926275
scientific article