I'm aware that the generator for one-dimensional geomtric BM is given by: $$\mathrm{A}f(x) = \mu xf'(x)+\frac{1}{2}\sigma^2x^2f''(x),$$ but I am trying to find the general infinitesimal generator for n dimensions.
I know that the general formula for the infinitesimal generator is given by $$\mathrm{A}f(x)=\sum_{i}b_i(x)\frac{\partial f}{\partial x_i}(x)+\frac{1}{2}\sum_{i,j}(\sigma\sigma^T)_{i,j}(x)\frac{\partial^2 f}{\partial x_i \partial x_j}(x)$$.
I suspect that the $\mu$ portion of the generator for the n-dimensional GBM will look similar to its one-dimensional counterpart, and my intuition says that the $\sigma$ portion will consist of the laplace operator, but I'm not sure of the steps to get there.
Thank you.