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 $\mathbb{R}<sup>d$, $d\ge 2$. Given a decomposition of the form [ \mathbf{G}=A\nablaΦ+\mathbf{B}, ] we relate the weighted divergence-free condition $\mathrm{div}_μ(\mathbf{B})=0$, where $μ=e<sup>{2Φ}dx$, to infinitesimal invariance of $μ$ 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.

• The paper establishes that a weighted divergence-free drift remainder is equivalent to an infinitesimally invariant measure, linking decomposition potentials, stationary Fokker–Planck equations, and Lyapunov functions.
• The paper characterizes Gaussian invariant measures for linear drifts through an algebraic Riccati equation and trace condition, proves existence for arbitrary drift matrices, and gives explicit formulas in two dimensions.
• The paper constructs verifiable polynomial perturbations with unique invariant measures, while counterexamples show that stability and uniqueness do not necessarily imply an orthogonal Helmholtz–Hodge decomposition.

Overview

This paper by Lee and Trutnau develops a unified analytic framework connecting three objects that are usually studied separately: Helmholtz–Hodge-type decompositions of drift vector fields, Lyapunov functions, and (infinitesimally) invariant measures for diffusion operators on $\mathbb{R}^d$, $d\ge 2$. The central object is the operator

$$L^{A,\mathbf{G}}f=\tfrac12\operatorname{trace}(A\nabla^2 f)+\langle \mathbf{G},\nabla f\rangle,$$

with $A$ symmetric positive definite. The key structural result is that if the drift admits a decomposition $\mathbf{G}=A\nabla\Phi+\mathbf{B}$ with weighted divergence-free remainder, i.e. $\mathrm{div}_\mu(\mathbf{B})=0$ for $\mu=e^{2\Phi}dx$, then $\mu$ is an infinitesimally invariant measure for $L^{A,\mathbf{G}}$, and conversely. This identifies the potential $\Phi$ directly as the logarithm of an invariant density candidate, so that constructing a decomposition is equivalent to solving the stationary Fokker–Planck equation in logarithmic form.

The paper distinguishes three levels of decomposition: the plain HHD ($d\ge 2$0), the orthogonal HHD (OHHD), which adds $d\ge 2$1 a.e., and the strictly orthogonal HHD (SOHHD), where these identities hold pointwise under $d\ge 2$2 regularity. OHHDs and SOHHDs imply WHHDs; geometrically, the gradient part moves trajectories normal to level sets of $d\ge 2$3 while $d\ge 2$4 flows tangentially, which yields the Lyapunov inequality $d\ge 2$5 automatically.

Uniqueness of the weighted decomposition

The main general uniqueness theorem states: if $d\ge 2$6 has a unique infinitesimally invariant measure $d\ge 2$7 (unique up to multiplicative constants), then $d\ge 2$8 is the unique WHHD; any OHHD, if it exists, coincides with it; and an OHHD exists if and only if either $d\ge 2$9 or $$L^{A,\mathbf{G}}f=\tfrac12\operatorname{trace}(A\nabla^2 f)+\langle \mathbf{G},\nabla f\rangle,$$0. The proof is short: any WHHD produces its own infinitesimally invariant measure via the equivalence theorem, so uniqueness of the measure forces equality of potentials up to constants, hence of gradients.

Uniqueness of the measure itself holds, e.g., when the associated Markov semigroup is recurrent, and the paper translates this into verifiable integral conditions involving $$L^{A,\mathbf{G}}f=\tfrac12\operatorname{trace}(A\nabla^2 f)+\langle \mathbf{G},\nabla f\rangle,$$1 and $$L^{A,\mathbf{G}}f=\tfrac12\operatorname{trace}(A\nabla^2 f)+\langle \mathbf{G},\nabla f\rangle,$$2: roughly, growth of $$L^{A,\mathbf{G}}f=\tfrac12\operatorname{trace}(A\nabla^2 f)+\langle \mathbf{G},\nabla f\rangle,$$3 slower than $$L^{A,\mathbf{G}}f=\tfrac12\operatorname{trace}(A\nabla^2 f)+\langle \mathbf{G},\nabla f\rangle,$$4 with $$L^{A,\mathbf{G}}f=\tfrac12\operatorname{trace}(A\nabla^2 f)+\langle \mathbf{G},\nabla f\rangle,$$5 suffices.

Two examples delimit the theory sharply. First, a linear drift $$L^{A,\mathbf{G}}f=\tfrac12\operatorname{trace}(A\nabla^2 f)+\langle \mathbf{G},\nabla f\rangle,$$6 with skew-symmetric quadratic part admits a unique WHHD with Gaussian $$L^{A,\mathbf{G}}f=\tfrac12\operatorname{trace}(A\nabla^2 f)+\langle \mathbf{G},\nabla f\rangle,$$7 and is recurrent/stable, yet admits no OHHD at all — stability plus uniqueness of the weighted decomposition do not force orthogonality. Second, constant or traceless linear fields can admit two distinct SOHHDs, showing nonuniqueness when the invariant measure is not unique (Lebesgue measure in those cases).

Linear drifts and the Riccati characterization

For $$L^{A,\mathbf{G}}f=\tfrac12\operatorname{trace}(A\nabla^2 f)+\langle \mathbf{G},\nabla f\rangle,$$8, Gaussian measures $$L^{A,\mathbf{G}}f=\tfrac12\operatorname{trace}(A\nabla^2 f)+\langle \mathbf{G},\nabla f\rangle,$$9 with symmetric $A$0 are characterized completely: $A$1 is infinitesimally invariant for $A$2 if and only if

$A$3

i.e., an algebraic Riccati equation together with a trace condition. For linear remainders $A$4, the WHHD condition collapses to the SOHHD condition because both reduce to pointwise identities.

Existence of a real symmetric solution $A$5 is proved for arbitrary $A$6 and symmetric positive definite $A$7 via a Schur-reordering argument: the spectrum is split into a conjugation-closed part containing $A$8-pairs (where $A$9 vanishes on the corresponding block) and a complementary part with no zero pair-sums, where the Lyapunov/vectorization method applies. In dimension $\mathbf{G}=A\nabla\Phi+\mathbf{B}$0, fully explicit closed-form formulas for $\mathbf{G}=A\nabla\Phi+\mathbf{B}$1 are derived for all cases (symmetric $\mathbf{G}=A\nabla\Phi+\mathbf{B}$2, zero-trace $\mathbf{G}=A\nabla\Phi+\mathbf{B}$3, singular rank-one $\mathbf{G}=A\nabla\Phi+\mathbf{B}$4, and nonsingular $\mathbf{G}=A\nabla\Phi+\mathbf{B}$5), extending Suda's formula from $\mathbf{G}=A\nabla\Phi+\mathbf{B}$6 to general $\mathbf{G}=A\nabla\Phi+\mathbf{B}$7 via Cholesky reduction.

In the Hurwitz case, the classical criterion is recovered structurally: the OU semigroup admits a finite invariant measure if and only if $\mathbf{G}=A\nabla\Phi+\mathbf{B}$8 is Hurwitz, with

$\mathbf{G}=A\nabla\Phi+\mathbf{B}$9

negative definite, unique, and yielding the unique SOHHD $\mathrm{div}_\mu(\mathbf{B})=0$0. Notably, the converse direction (finite invariant measure implies Hurwitz) is obtained through the Riccati characterization rather than spectral analysis of the semigroup. The appendix demonstrates concretely that the Lyapunov-equation route fails precisely when eigenvalue pair-sums vanish (e.g., $\mathrm{div}_\mu(\mathbf{B})=0$1 with spectrum $\mathrm{div}_\mu(\mathbf{B})=0$2, or a pure rotation), while the direct algebraic construction still succeeds — a genuine extension beyond the Hurwitz regime.

Nonlinear polynomial perturbations

The nonlinear section asks when a Gaussian-derived potential survives perturbation by higher-order polynomial drifts. The main theorem gives sufficient conditions: if $\mathrm{div}_\mu(\mathbf{B})=0$3 is an even-degree polynomial satisfying a tail bound $\mathrm{div}_\mu(\mathbf{B})=0$4 with $\mathrm{div}_\mu(\mathbf{B})=0$5, and the perturbation satisfies the algebraic identity $\mathrm{div}_\mu(\mathbf{B})=0$6, plus a Lyapunov-type bound $\mathrm{div}_\mu(\mathbf{B})=0$7 off a large ball for $\mathrm{div}_\mu(\mathbf{B})=0$8, then $\mathrm{div}_\mu(\mathbf{B})=0$9 is finite and is the unique finite invariant and infinitesimally invariant measure, and the induced decomposition is conservative. A checkable criterion reduces these hypotheses to: $\mu=e^{2\Phi}dx$0 and strict negativity of the leading homogeneous part $\mu=e^{2\Phi}dx$1 on the sphere. The paper explicitly notes that the sphere condition is sufficient but not necessary — $\mu=e^{2\Phi}dx$2 shows that $\mu=e^{2\Phi}dx$3 does not imply it, and direction-dependent arguments would be needed there.

In $\mu=e^{2\Phi}dx$4 with quadratic perturbations, the invariance conditions become an explicit linear system in the perturbation coefficients, and the dichotomy is sharp: if the divergence-free conditions on the quadratic part fail, no SOHHD exists; if they hold, the decomposition is the unique OHHD. A worked example with Hurwitz $\mu=e^{2\Phi}dx$5 and nontrivial antisymmetric quadratic perturbation exhibits a unique Gaussian invariant measure preserved under a genuinely non-reversible, non-orthogonally-decomposable drift.

A separate mechanism uses skew-symmetric matrices of functions: for $\mu=e^{2\Phi}dx$6 with $\mu=e^{2\Phi}dx$7 antisymmetric, the decomposition with potential $\mu=e^{2\Phi}dx$8 is always a WHHD, becomes an OHHD whenever $\mu=e^{2\Phi}dx$9, and under Hurwitzness of $\mu$0 the resulting family $\mu$1 carries $\mu$2 as its unique invariant measure for all triples $\mu$3. Concrete admissible classes include $\mu$4 with constant antisymmetric $\mu$5; in $\mu$6 the polynomial analysis shows this is essentially the only quadratic choice compatible with the orthogonality constraint.

Limitations and open questions

Several restrictions are acknowledged explicitly. The existence theorem for closed-form solutions of the Riccati system is complete only for $\mu$7; for $\mu$8 the authors state it is unclear whether a closed form exists, and the general existence proof is constructive only through Schur decompositions and Kronecker-system inversion. The Lyapunov criterion via the leading homogeneous part excludes potentials like $\mu$9 whose leading term degenerates on the sphere, and no direction-dependent substitute is developed. The recurrence-based uniqueness criteria for WHHDs rely on prior results of Lee–Stannat–Trutnau and Gim–Trutnau rather than self-contained proofs. Finally, the framework treats only diffusion operators with constant diffusion matrix $L^{A,\mathbf{G}}$0 on Euclidean space; extensions to degenerate or position-dependent $L^{A,\mathbf{G}}$1, or to manifolds, are outside the scope of the results proved here.

Conclusion

The paper establishes that weighted Helmholtz–Hodge decompositions provide a common algebraic language for Lyapunov functions and invariant measures of non-reversible diffusions: the weighted divergence-free condition is exactly infinitesimal invariance, uniqueness of the invariant measure canonizes the potential, and the Riccati-plus-trace system governs the Gaussian case, including a structural rederivation of the Hurwitz criterion. The explicit polynomial perturbation classes constructed here give verifiable instances of non-reversible dynamics with prescribed, explicitly known invariant measures, while the counterexamples delineate precisely when orthogonal structure fails despite stability and uniqueness.