TITLE: What is the importance of the infinitesimal generator of Brownian motion?
QUESTION [72 upvotes]: I have read that the infinitesimal generator of Brownian motion is $\frac{1}{2}\small\triangle$. Unfortunately, I have no background in semigroup theory, and the expositions of semigroup theory I have found lack any motivation or intuition.
What is the infinitesimal generator of a process intuitively, and why is it interesting or useful to know that the generator of Brownian motion is $\frac{1}{2}\small\triangle$?

REPLY [17 votes]: In fact, there is a deeper relation between the Laplacian and Brownian motion.
Let $(M, g=\langle\cdot, \cdot\rangle)$ be a smooth Riemannian manifold without boundary. The Laplace-Beltrami operator is defined as the contraction of the covariant derivative of the differential of any smooth function on $M$
$$\forall f \in C^\infty(M): \Delta_M f := \mathrm{tr} \nabla \mathbf df = \mathrm{div}\ \mathrm{grad} \ f \in C^\infty(M),$$
where the well-known definition can be recovered with suitable generalisations of the divergence and the gradient. This means, for any orthonormal basis $E_1,...E_n$ for $T_pM$ ($p \in M$),
$$\forall f \in C^\infty(M): \Delta_M f(p) = \sum_{i=1}^n \nabla\mathbf d f(E_i,E_i) = \left\langle \nabla_{E_i}\mathrm{grad} \ f, E^i \right\rangle,$$
where we used Einstein notation. Moreover, we can generalise the term of a continuous semimartingale as follows: Every adapted $M$-valued stochastic process $X$ is a semimartingale on $M$ if, for all $f \in C^\infty(M)$, the composition map is $f(X)$ a real-valued semimartingale.
Then we can define Brownian motion on $M$ by the usual martingale problem (this is known as the extrinsic definition):
Let $X$ an adapted $M$-valued process. A process $X$ is called Brownian motion on $(M,g)$ if, for all $f \in C^\infty(M)$, the real-valued process
$$f(X) - \frac 12 \int \Delta_M f(X) \mathrm dt$$
is a local martingale.
In particular, we can prove Lévy's characterisation also for BM$(M,g)$. But this requires a reasonable definition of the quadratic variation.
The problem with this definition lies in the manifold itself: There does not exist a Hörmander-type representation of the Laplace-Beltrami operator if $M$ is not parallelizable, i.e. the tangent bundle $TM \overset\pi\longrightarrow M$ is not trivial. But it holds the fundamental relation
$$\Delta_{\mathcal O(M)} \pi^* = \pi^* \Delta_M,$$
more precisely,
$$\Delta_{\mathcal O(M)}(f \circ \pi)(u) = \Delta_M f(x),$$
for all $u \in \mathcal O(M)$ with $x = \pi(u)$. Moreover, there exist $n$ well-defined unique horizontal vectors $L_i(u) \in H_u\mathcal O(M)$, $\pi_* L_i(u) = ue_i$, $(e_i)$ basis for $\mathbb R^n$, the so called fundamental horizontal vector fields and the we define
$$\Delta_{\mathcal O(M)} := \sum_{i=1}^n L_i^2,$$
where $\mathcal O(M)$ denotes the orthonormal frame bundle, the prototypical example of a smooth principal fibre bundle whose structure group is given by the orthogonal group.
Using this relation, it is due to Malliavin, Eells and Elworthy that there always exists a lifted Brownian motion as solution of the globally defined SDE
$$\mathrm d U = L_i(U) \circ \mathrm d B^i,$$
on $\mathcal O(M)$, where $B$ is a real $n$-dimensional Brownian motion and we used Einstein notation. A solution is a diffusion generated by $\frac 12\Delta_{\mathcal O(M)}$. The idea is to solve the SDE in $\mathcal O(M)$ and $X = \pi(U)$ is the projection of the lifted Brownian motion $U$ on the manifold $M$ via $\mathcal O(M) \overset\pi\longrightarrow M$. It follows that $X$ is a Brownian motion on $M$ starting from $X_0 = \pi(U_0)$.
In geometrical terms, the idea is to "roll" our manifold $M$ by means of the (stochastic) parallel displacement along the paths of an $\mathbb R^n$-valued Brownian motion ("rolling without slipping"), known as stochastic development.
References:

Hsu, Elton P. Stochastic analysis on manifolds. Vol. 38. American Mathematical Soc., 2002.
(in german) Hackenbroch, Wolfgang, and Anton Thalmaier. Stochastische Analysis. Vieweg+ Teubner Verlag, 1994.
Elworthy, Kenneth David. Stochastic differential equations on manifolds. Vol. 70. Cambridge University Press, 1982.
Malliavin, Paul. Géométrie différentielle stochastique. Montreal, Presses de l’universite de Montreal, 1978.