Infinitesimal generator of the Brownian motion on a circle

Question

As explained here, the infinitesimal generator of a Brownian motion is $\frac{1}{2}\Delta$. Can someone please provide (for a non-stochastic-student) the proof of finding the infinitesimal generator on a circle (and if possible, also on a sphere) in the Cartesian coordinates? (or introducing a book or article)

Thanks.

Answer 1

Starting from $$dY_1=-\frac12 Y_1 dt-Y_2 dB_t, \\ dY_2=-\frac12 Y_2 dt+Y_1 dB_t, $$

and using the general formula for the generator of a diffusion process $dX_t=f(X_t)dt+g(X_t)dB_t$, which reads $$\mathcal{A}=\sum_{i=1}^n f_i(x)\partial_{x_i}+\frac{1}{2}\sum_{i=1}^n\sum_{j=1}^m g_{ij}(x)\partial_{x_i}\partial_{x_j},$$ we have with $n=2$ and $m=1$ $$\mathcal{A}=\frac12\left(-y_1\partial_{y_1}-y_2\partial_{y_2}+y_2^2\partial_{y_1}^2+y_1^2\partial_{y_2}^2\right)=\frac12\partial_{\theta}^ 2.$$ Which is one half times the Laplacian on $S_1$.

In fact, one of the definitions of Brownian motion on a Riemannian manifold $(M,g)$ is that its generator is $\frac12\Delta_g$, where $\Delta_g$ is the Laplacian of $g$. Thus in order to find the generator of Brownian motion on $S_2$ in Cartesian coordinates, take the Laplacian on $S_2$ and transform it to Cartesian coordinates (a slightly tedious calculation).