Defining Brownian Motion on a Sphere Through Its Generator
Summary
The document contrasts Brownian motion in Euclidean space with Brownian motion on a sphere. In Euclidean space, familiar characterizations use the vector-space structure and Gaussian increments. A sphere lacks that structure, so those characterizations do not directly define a process on it. As a Riemannian manifold, however, the sphere has a natural Laplacian, which supplies a way to define Brownian motion as a continuous-time Markov process with generator one half of that operator.
The answer also describes a constructive route: form random walks on the sphere, then interpolate and rescale them. Under an appropriate invariance principle, these walks converge to a continuous-time Markov process with the same generator. This gives mathematical intuition for defining the process through an operator or obtaining it as a limit. The discussion is conceptual and does not detail the convergence conditions or a physical model for particles moving on a sphere.
Key ideas
- Euclidean Brownian motion can be characterized using vector-space and Gaussian structure.
- A sphere’s manifold structure provides a natural Laplacian for defining Brownian motion.
- The process can be specified as a continuous-time Markov process with generator one half the Laplacian.
- Rescaled random walks on the sphere can converge to a process with that generator.
- The document omits technical convergence conditions and a detailed physical interpretation.
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# Infinitesimal generator - Is it obtained from a stochastic process or It can construct the process
# Infinitesimal generator - Is it obtained from a stochastic process or It can construct the process
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 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 $S^n$?
## Answer by Tobsn (score 1)
https://quant.stackexchange.com/a/57616
The problem simply is that $S^{n}$ is not a group or a Hilbert space. Therefore you cannot make sense of BM on $S^{n}$ via Levy-characterisation or as a Gaussian process. However, we know $S^{n}$ is a Riemannian manifold and it comes along with a natural notion of the Laplacian. Therefore, one of the easiest ways to define BM on $S^{n}$ is by postulating it to be the continuous time Markov process with generator $\frac{1}{2}\Delta$. But that's not the only way to do. Another way would e.g. be via approximation. Even without a group structure, one can make sense of random walks on $S^{n}$. Then akin to the argument in $\mathbb{R}^{n}$, which goes under Donsker's invariance principle or functional central limit theorem, one can show that after appropriate interpolation and rescaling this random walk will converge to some time continuous Markov process, whose generator is indeed again $\frac{1}{2}\Delta$.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.