Let \(\Delta\) be the Laplace operator on \(S^d\), acting on vector fields. The spectrum of \(\Delta\) is given by \(\{-c_{l,d}:l\geq 1\}\cup\{-c_{l,\delta}:l\geq 1\}\), where \(c_{l,d}=l(l+d-1)\), \(c_{l,\delta}=(l+1) (l+d-2)\). Let \(G_l\) and \(D_l\) be the eigenspaces associated to \(c_{l,d}\) and \(c_{l,\delta}\), respectively, \(D_{l,1}=\dim G_l\), \(D_{l,2}=\dim D_l\). By \(\{A^i_{l,k}:k=1,\dots,D_{l,1},l\geq 1\}\), \(i=1,2\), the orthonormal bases of \(G_l\) and \(D_l\) in \(L^2\) are denoted. Let \(a_l=\frac {a}{(l-1)^{1+\alpha}}\), \(b_l=\frac{b}{(l-1)^{1+\alpha}}\), \(\alpha>0\), \(a,b>0\), \(l\geq 2\) and \(\{B^i_{l,k}:l\geq 1,1\leq k\leq D_{l,i}\}\), \(i=1,2\), be two families of independent standard Brownian motions defined on a probability space \((\Omega,F,P)\). For \(\alpha=2\) the authors consider the Stratonovich stochastic differential equation on \(S^d\):
\[ dx_t^n= \sum^{2^n}_{l=1}\left\{\sqrt{\frac {da_l}{D_{l,1}}} \sum^{D_{l,1}}_{k=1}A^1_{l,k}(x_t^n)\circ dB^1_{l,k}(t)+\sqrt {\frac{db_l}{D_{l,2}}}\sum^{D_{l,2}}_{k=1}A^2_{l,k}(x^n_t)\circ dB^2_{l,k}(t) \right\},\quad x_0^n=x. \]
Using the specific properties of eigenvector fields, they prove that \(x_t^n(x)\) convergens uniformly in \((t,x)\in[0,T]\times S^d\). The uniform limit \(x_t(x)\), \(t\geq 0\), is the unique solution of the equation \[ dx_t= \sum^\infty_{l=1}\left\{\sqrt {\frac{da_l}{D_{l,1}}}\sum^{D_{l,1}}_{k=1}A^1_{l,k}(x_t)\circ dB^1_{l,k}(t)+\sqrt{\frac{db_l}{D_{l,2}}}\sum^{D_{l,2}}_{k=1} A^2_{l,k} (x_t)\circ dB^2_{l,k}(t)\right\},\quad x_0=x, \]
which gives rise to a flow of homeomorphisms.