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

Abstract

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 \[ \mathbf{G}=A\nabla\Phi+\mathbf{B}, \] we relate the weighted divergence-free condition $\mathrm{div}_{\mu}(\mathbf{B})=0$, where $\mu=e^{2\Phi}dx$, to infinitesimal invariance of $\mu$ for the operator \[ \frac12 \mathrm{trace}(A\nabla^2)+\langle \mathbf{G},\nabla\cdot\rangle. \] 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.