Let $M$ be a manifold defined by an implicit equation $f(x)=0$ for some smooth function $f:\mathbb{R}^D\to \mathbb{R}^K$, where $K=D-d$. Assume the Jacobian matrix $Df(x)$ has rank $K$ on $M$. Brownian motion can be written as the Stratonovich SDE $$dX_t = P(X_t) \circ dW_t,$$ where $W$ is a $D$-dimensional standard Brownian motion and $P(x) = I-N(x)^TN(x)$ where $N$ is obtained from orthonormalizing the rows of $Df$. When the manifold is a hypersurface, it is easy to compute the It^{o} drift term, it is $c(x) \vec{n}(x)$ where $c(x)=-\frac12 \operatorname{div}(n(x))$ is the mean-curvature of the hypersurface at $x$ and $\vec{n}(x)$ is the normal vector of the surface at $x$, given by $\vec{n}(x)=\nabla f(x)/\|\nabla f(x)\|$.
Thus, when $K=1$, the geometric intuition of the Euler-Maruyama 1-step is easy to interpret; the step $$x_h =x_0 +c(x_0)n(x_0)h+P(x_0)\sqrt{h}Z$$ can be interpreted as follows. Watch some noise starting at the origin, and stop it after time $h$: we see $\sqrt{h}Z$. Orthogonally project this onto the tangent space of $M$ at $x_0$, we get $P(x_0)\sqrt{h}Z$. Now displace $x_0$ by this vector--this means we travel along the tangent hyperplane at $x_0$. Then we displace once more in the normal direction to $M$ at $x_0$ from this point along the tangent hyperplane--the magnitude of the displacement being exactly the mean curvature of the manifold--i.e. the local/infinitesimal displacement from the tangent hyperplane. This is quite easy to draw in the plane when dealing with $M = $ a curve, and if you are good at 3D drawings, also surfaces in $\mathbb{R}^3$.
Question: How do we interpret the same motion when $M$ is an intersection of hypersurfaces? For example, Brownian motion on a space-curve (so $D=3$ and $d=1$, $K=2$). Previously, I have asked about what the drift term is in such a case, in general, and have derived a formula that is not easy for me to interpret geometrically: see this question. In doing examples, I have seen its something like $N^T(x) c(x)$ where $c$ is now a vector of mean-curvatures, in the direction of the $K$ normals.
I had an idea that we could consider each hypersurface separately, and run the same gamut as before and then try to bring it all together back in the total space. So you would project down to say the $xy$ plane and $xz$ plane, displace along the tangents, project normally down to $M$ a magnitude that is equal to that mean-curvature and then bring it back up. Is any of this on the right path?
I apologize for any vagueness, mistakes, or missing obvious stuff--still trying to recover lost geometric intuition from high school...
