We can see here that the generator is an operator which can be determined for a stochastic process. But, in the answers and comments here we can see that the brownian motion on sphere(or manifolds) can be constructed by the assumption that the generator is $\frac{1}{2}\Delta$.
Can anybody explain (for a novice in stochastic calculus), why can we "find" a generator for a Brownian motion in $R^n$, whereas, for the Brownian motion on sphere, first we assume something and then we construct the BM based on that generator? What is intuitively/physically/mathematically the difference between them?
What is the physics of the standard Brownian motion on a sphere? In $R^n$, i think, the formulation corresponds to the random motion of particles in a fluid. What about BM on $S^n$?