Infinitesimal generator of brownian motion

Question

It the literature we see that the infinitesimal generator of Brownian motion is $\frac{1}{2} \Delta$. However, when we search about the generator of the Brownian motion on the (unit) sphere, we see that the brownian motion is constructed to have the generator of $\frac{1}{2} \Delta_{S^2}$. Why the language is different in these two cases? In $\mathbb R^n$, the generator is calculated from the process. On sphere, the process is defined such that it has generator $\frac{1}{2} \Delta_{S^2}$. How can it be explained?

Answer 1

The difference comes from the different spaces they exist in.

Standard Brownian motion is on flat, Euclidean space and its generator is the Laplacian $\frac{1}{2}\Delta$ (Laplacian is a differential operator that measures the divergence of the gradient of a function).

When we move to a curved space like the sphere the Laplacian is adjusted to $\Delta_{s^2}$ to account for the curvature, and the generator of the Brownian motion on the sphere becomes $\frac{1}{2}\Delta_{s^2}$.

The generator of the Brownian motion whether in flat space or on a sphere represents the underlying dynamics of the motion. The difference in the generators comes from the difference in the spaces where the motions are defined.

I believe the proof for the generator of standard Brownian motion is relatively straightforward. It is derived from the definition of Brownian motion and the properties of the Laplacian operator.

In the 1D case it can be showed that $Af(x) := \lim_{t\rightarrow 0} \frac{\mathbb{E}^x[f(B_t)] - f(x)}{t} = \frac{1}{2}f''(x)$.

Taylor expansion about $f(x)$:

\begin{align} \mathbb{E}^x[f(B_t)] &\approx \mathbb{E}^x[ f(x) + (B_t-x) f'(x) + \frac{1}{2}(B_t-x)^2 f''(x)]\ &= f(x) + \mathbb{E}^x[B_t-x] f'(x) + \frac{1}{2} \mathbb{E}^x[(B_t-x)^2] f''(x)\ &= f(x) + \frac{t}{2} f''(x) \end{align}

This result shows that the generator of a standard Brownian motion is indeed $1/2 \Delta$

For the Brownian motion on a sphere, the proof is more involved and requires the use of stochastic differential equations (SDEs) in spherical coordinates.

The generator of Brownian motion on $S^2$ (with the round metric) is $\frac12\Delta$, where $\Delta$ is the Laplacian on $S^2$, in spherical coordinates $$\Delta = \frac{1}{\sin\theta}\partial_{\theta}(\sin\theta\cdot\partial_{\theta})+\frac{1}{\sin^2\theta}\partial_{\varphi}^2.$$

The Brownian motion on $S^2$ is then defined by the following system of SDEs:

\begin{align} d\theta_t&=\frac12\cot\theta_t dt+\sin\varphi_t dB_t^{(1)}-\cos\varphi_t dB_t^{(2)},\ d\varphi_t&=\cot\theta_t\left(\cos\varphi_t dB_t^{(1)}+\sin\varphi_t dB_t^{(2)}\right)-dB_t^{(3)}. \end{align}

Using Itô's lemma, one can then check that for a $C^2$ function $f(\theta_t,\varphi_t)$:

$$df(\theta_t,\varphi_t) = \frac12\Delta f(\theta_t,\varphi_t) dt + ... dB_t^{(1)}+ ... dB_t^{(2)}+ ... dB_t^{(3)}$$

$dt$ term is $\frac{1}{2}\Delta f(\theta_t,\varphi_t)$ which shows that the generator of the process is indeed $\frac{1}{2}\Delta$.