Consider a Lie group \(G\), a compact Lie subgroup \(H\), their Lie algebras \(\mathfrak g\), \(h\) respectively, an \(\mathrm{Ad}_H\)-invariant Lie subalgebra \(\mathfrak{m}\) such that \(\mathfrak{g} = h\oplus m\), a generating subset \(\{A_0,\ldots,A_N\}\subset \mathfrak{h}\), and \(\,Y_0\in \mathfrak{m}\). Then the canonical projection \(\pi : G\to G/H\) makes \(G\) into a principal bundle over the homogeneous manifold \(M:=G/H\).
Denote by \((b^1_t,\ldots,b_t^N)\) an \(N\)-dimensional Euclidean Brownian motion. For any small positive \(\varepsilon\), let \((g_t^\varepsilon)\) be the solution to the Stratonovitch SDE
\[ g_t^\varepsilon = g_0 + \varepsilon^{-1/2}\sum_{k=1}^N\int_0^t A_k(g_s^\varepsilon)\circ db_s^k +\varepsilon^{-1}\!\int_0^t A_0(g_s^\varepsilon)\, ds + \int_0^t Y_0(g_s^\varepsilon)\, ds, \] where as usual any \(V\in \mathfrak{g}\) is identified with the left-invariant vector field it generates. Thus, \((g_t^\varepsilon)\) is a diffusion with generator \(\varepsilon^{-1}\mathcal{L}_0+Y_0\), where \(\frac{\mathcal L_0 := 1}{2\sum_{k=1}^NA_k^2+A_0}\). No Hörmander condition is required. Let \(x_t^\varepsilon:= \pi(g_t^\varepsilon)\).
The author considers the lift \(u_t^\varepsilon\in G\) above \(x_t^\varepsilon\), which be horizontal with respect to the Ehresmann connection determined by \(\mathfrak{m}\), and also the vertical semimartingale \((u_t^\varepsilon)^{-1} g_t^\varepsilon\in H\). She computes SDE’s governing both processes.
Her main result is the weak convergence, as \(\varepsilon\searrow 0\), of the scaled semimartingale \((u_{t/\varepsilon}^\varepsilon)\), towards some diffusion process, with an explicit expression for the limiting generator.
Moreover, this convergence is proved to hold in Wasserstein distance as well, and a rate of convergence is given. Furthermore, conditions are given which ensure that the limiting diffusion is a scaled Brownian motion, with some computable scale. Finally, some examples are discussed.