Identify an SDE on the sphere from its generator

Question

I have a diffusion on the 2-sphere with expression:

$$ (L\phi)(u):=\frac{1}{2{N(u)}}\Big(f(u)\Delta_{\mathbb S^2}\phi+ 2g\left( \nabla_{\mathbb S^2}\phi, \nabla_{\mathbb S^2}f\right)\Big) $$ for some positive functions $N,f$. Here, $\Delta_{\mathbb S^2}$ is the Laplace-Beltrami operator of the sphere and $g$ is the induced metric on the sphere. I would like to find a SDE possibly in the ambiant space $\mathbb R^3$ such that $L$ is its generator.

From Watanabe p288, I could also change the metric and connections to construct a process on the frame bundle but this seems tedious.

I know (see Hsu) that in the case $L_0=\Delta_{\mathbb S^2}$, the process can be constructed by embedding in $\mathbb R^3$ (aka the Stoock's representation of spherical brownian motion). In the case $\frac{f(u)}{g(u)}\Delta_{\mathbb S^2}$, I am not sure how to proceed already.

Answer 1

If you are happy with any SDE (denoted i) after) in $\mathbb R ^d$ with associated second order generator $L$, you can safely derive a perturbed SDE from it with associated generator $ a \times ( L + b^i \partial_i ) $ in which $ a> 0$ is any positive scalar field and $b$ any Lipschitz vector field.

To do so, you just need to add the vector field $b$ in the SDE i) and then perform a time change of scale $a(X_t)$ ($X_t$ is the solution of the SDE). This precisely amounts to multiply in the SDE i) each $d t$ term by $a(X_t)$ and each $d W_t$ Itô term by $a^{1/2}(X_t)$.

In particular, if the initial SDE i) belongs almost surely to a sub-manifold of $\mathbb R^d$ and if $b$ is tangent to this submanifold, then the perturbed SDE will again almost surely belong to this submanifold.

Note that you can change drift and conformal factor of the metric by this method but not non-scale features of the (perhaps induced) metric itself.

(To be clear if i) is $d X_t = f( X_t ) d t + \sigma(X_t) d W_t$ with generator $L = b^i \partial_i + \frac12 (\sigma \sigma^\dagger )^{i,j} \partial_i\partial_j$ then the here discussed perturbed SDE is $ d X_t = a(X_t) ( f( X_t ) + b( X_t) ) d t + a^{1/2}(X_t)\sigma(X_t) d W_t$ with generator $ aL + a <b,d \, . \, > $.)

I think you can easily construct the SDE associated with you generator starting from any SDE in $\mathbb R ^3$ generating a BM on $ \mathbb S ^2$ in this elementary way.

Note that it can be useful also to use Stratonovitch notation in those cases because the associated chain rule enables to write geometrically coordinate invariant SDEs. For instance if $P$ denotes the orthogonal projector onto the tangent space of a submanifold then the SDE: $$ d X_t = P(X_t) \circ d W_t $$ where $W_t$ is $d$-dimensional is automatically a geometric BM of the sub-manifold.

Be careful with time change in this case, you need to develop the Stratonovitch integration into a Itô one in order to get the correct formula under time change. But at the level of generators, you can obtain easily geometrically intrinsic formulae (Nota Bene: For geometric BM, these types of time change calculus is equivalent to the geometric formula yielding the Laplace operator under a conformal transformation of the underlying metric. A symptom of that is that the projector is invariant by conformal transformation, but the Laplace operator isn't).