Weighted Helmholtz--Hodge decompositions, Lyapunov functions, and invariant measures

We study weighted Helmholtz–Hodge decompositions of drift vector fields associated with second-order diffusion operators on ${\mathbb{R}}^d$, $d\ge 2$. Given a decomposition of the form $$\mathbf{G}=A\mathrm{\nabla }\mathrm{\Phi }+\mathbf{B},$$ we relate the weighted divergence-free condition ${\mathrm{d}\mathrm{i}\mathrm{v}}_{\mu }(\mathbf{B})=0$, where $\mu =e^{2\mathrm{\Phi }}dx$, to infinitesimal invariance of $\mu$ for the operator $$\frac{1}{2}\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}(A{\mathrm{\nabla }}^2)+⟨\mathbf{G},\mathrm{\nabla }\cdot ⟩\mathrm{.}$$ We compare weighted, orthogonal, and strictly orthogonal Helmholtz–Hodge decompositions and show that uniqueness of the infinitesimally invariant measure yields uniqueness of the corresponding weighted decomposition, and hence a canonical potential. For linear vector fields, we characterize Gaussian infinitesimally invariant measures by an algebraic Riccati equation together with a trace condition. In the Ornstein–Uhlenbeck case, this gives a structural proof of the classical criterion that a finite invariant measure exists if and only if the drift matrix is Hurwitz, and it identifies the associated strictly orthogonal decomposition. Finally, we treat nonlinear polynomial perturbations that preserve a given potential and obtain explicit classes of drifts for which the invariant measure and the weighted decomposition remain unique. The results clarify the relation between Lyapunov-type potentials, non-reversible perturbations, and invariant measures for diffusion semigroups.