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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 $S^n$?

Denis
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  • Why shouldn't it be both? – Nate Eldredge Aug 19 '20 at 16:46
  • The member in the linked question argued that, if we don't know the generator, how can we construct the BM on sphere? It seems that the existence of BM can be checked only by a pre-known generator... But i have no idea, i am still confused – Denis Aug 19 '20 at 16:50
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    Well, one has to have a definition of "Brownian motion on the sphere" before one can talk about whether it exists. If you define it as the process having the Laplacian as its generator then of course you can't avoid it. But there are other equivalent definitions, and one can construct a process satisfying those definitions without talking about the Laplacian. For example, take a standard Brownian motion $B_t$ on $\mathbb{R}^{n+1}$ and consider $B_t/|B_t|$. – Nate Eldredge Aug 19 '20 at 17:53
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    You may think of Brownian motion on $S^n$ as also being the random motion of particles in a fluid, except that the fluid is forming a thin film on the surface of the sphere and the particles are constrained to stay within the fluid. – Nate Eldredge Aug 19 '20 at 17:55
  • Good points, thank you. About taking the BM in $R$ and then the mapping, do you know any book/article which rigorously talk about the mapping from $R$ to $S$ in the context of the brownian motion? I also also similar question here. So if you had any hint, it is appreciated... – Denis Aug 19 '20 at 18:30
  • I'm afraid I cannot take the time to look one up at present. Sorry. – Nate Eldredge Aug 19 '20 at 18:58

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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?

To be clear even in $\mathbb{R}^{n}$, one again needs to start from some generator and then define the corresponding semigroup (satisfying the Cauchy-problem) and stochastic process. For Brownian motion we picked that particular transition density precisely because it satisfies the heat equation.

Once some basic processes were built, then this procedure was back-engineered too eg. see Feynman-Kac, of the correspondence, so one can start from an SDE and figure out the corresponding generator.

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 $S^n$?

For the math picture, take a look at Examples 3.3.2./3.3.3. in "Stochastic Analysis On Manifolds" by Elton P. Hsu. He derives the generator for Brownian motion on the sphere.

For a possible physics-picture, see "Brownian self-driven particles on the surface of a sphere" or "A note on the exact simulation of spherical Brownian motion". For example, one cool application is bacteria on the surface of cells

In applications one is often interested in the BM on curved surfaces and other manifolds, see e.g. Krishna et al. (2000) and Li et al. (2008) for the modelling of the fluorescent marker molecules in cell membranes and the motion of bacteria or any other diffusing particles, respectively.

Thomas Kojar
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