I have a diffusion on the 2-sphere with expression:

$$
(L\phi)(u):=\frac{1}{2{N(u)}}\Big(f(u)\Delta_{\mathbb S^2}\phi+
		2g\left( \nabla_{\mathbb S^2}\phi, \nabla_{\mathbb S^2}f\right)\Big)
$$
for some positive functions $N,f$. Here, $\Delta_{\mathbb S^2}$ is the Laplace-Beltrami operator of the sphere and $g$ is the induced metric on the sphere. I would like to find a SDE possibly in the ambiant space $\mathbb R^3$ such that $L$ is its generator.

From Watanabe p288, I could also change the metric and connections to construct a process on the frame bundle but this seems tedious.

I know (see Hsu) that in the case $L_0=\Delta_{\mathbb S^2}$, the process can be constructed by embedding in $\mathbb R^3$ (aka the Stoock's representation of spherical brownian motion). In the case $\frac{f(u)}{g(u)}\Delta_{\mathbb S^2}$, I am not sure how to proceed already.