Consider first a rescaled brownian motion $X_t$, in $\mathbb{R}^n$ which fulfills the SDE
$$dX_t = \sqrt{2} dB_t,$$
where $B_t$ is brownian motion. Then the density $p(t,x,x_0)$ of the process fulfills the Fokker-Planck equation $$\partial_t p = \Delta p$$ where $$\Delta = \nabla^* \nabla$$ is the Laplacian (and $\nabla$ is the gradient, $\nabla^*$ its adjoint).
This relation between heat equation and SDEs is typically taught in an stochastic analysis course. This made me automatically assume that there is a more general relation between FPE and the heat equation, but now I am struggling to see it exactly.
The general anisotropic heat equation reads
$$\partial_t p = \sum_{i,j} \partial_i (A_{ij}(x) \partial_j p_t(x)) =: \Delta_A p$$
where $A: \mathbb{R}^n \to \mathbb{R}^{n \times n}$ is a symmetric, positive definite matrix at all points. $\Delta_A$ can again be formulated as a self-adjoint operator, by considering the scalar-product given by $$\langle f, g \rangle_A = \int f(x)^TA(x)g(x)dx.$$ With this scalar product, and the standard scalar product on the scalar-functions, we then have $$ \Delta_A = \nabla^* \nabla,$$ in equivalence to the standard heat equation.
I then thought the corresponding SDE should be
$$ dX_x = \sqrt{2A(X_s)}dB_s,$$
but this then yields
$$ \partial_t p = \sum_{i,j} \partial_i \partial_j (A_{ij}(x) p_t(x)) =: L_A.$$
We see that $L_A \neq \Delta_A$ so this does not seem to be the correct equation (apart from the constant case, where both coincide).
However I also do not see how this could be salvaged, i.e. if there is another SDE that would produce the correct form. If this is in general not possible, then I would also be glad about some intuition why the connection between heat equation and SDEs holds only for the constant case.
Edit Following @Ian's comment, we can obtain a solution by including a drift term. We have
$$ \sum_{ij} \partial_i \partial_j (A_{ij} p) = \sum_{i,j} \partial_i (\partial_j A_{ij}) p + A_{ij} \partial_j p) = \sum_{i,j} \partial_i \partial_j A_{ij} p + \partial_i A_{ij} \partial_j p + \partial_j A_{ij} \partial_i p + \partial_i \partial_j p A_{ij} $$
and $$\sum_{i,j} \partial_i (A_{ij} \partial_j p) = \sum_{i,j} \partial_i A_{ij} \partial_j p + A_{ij} \partial_i \partial_j p$$
Such that when we consider the equation
$$dX_t = \mu_A(X_t) d_t + \sqrt{2A}dB_t$$
where $\mu_A(x)_i = \sum_j \partial_j A_{ij} (x)$, then for this process we have $$\partial_t p = \Delta_A p.$$
So for the interpretation I guess we have to imagine particles being pushed into the direction where there is more diffusivity i.e. better heat conduction