Consider the linear $d$-dimensional SDE
$$dX_t=-AX_tdt+\sqrt{2}BdW_t$$
for $X_t\in\mathbb{R}^d,A\in \mathbb{R}^{d\times d}$, and an invertible matrix $B\in \mathbb{R}^{d\times d}$, here $W_t$ is a $d$-dimensional Brownian motion. Its associated Fokker-Planck equation for the evolution of its density $\rho$ is
$$\partial_t\rho=\text{div}(Ax\rho+D\nabla\rho),~D:=BB^T.$$
Are there any distinguishing features for these linear SDE when we have the following commuting of matrices $AD=DA$ ?