Weighted Helmholtz--Hodge decompositions, Lyapunov functions, and invariant measures
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arXiv
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2026
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| author | Lee, Haesung Trutnau, Gerald |
| author_facet | Lee, Haesung Trutnau, Gerald |
| contents | 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 \[
\mathbf{G}=A\nablaΦ+\mathbf{B}, \] we relate the weighted divergence-free condition $\mathrm{div}_μ(\mathbf{B})=0$, where $μ=e^{2Φ}dx$, 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_25715 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Weighted Helmholtz--Hodge decompositions, Lyapunov functions, and invariant measures Lee, Haesung Trutnau, Gerald Probability Primary 60J35, Secondary 47D07, 60J60, 35Q84, 15A24 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 \[ \mathbf{G}=A\nablaΦ+\mathbf{B}, \] we relate the weighted divergence-free condition $\mathrm{div}_μ(\mathbf{B})=0$, where $μ=e^{2Φ}dx$, 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. |
| title | Weighted Helmholtz--Hodge decompositions, Lyapunov functions, and invariant measures |
| topic | Probability Primary 60J35, Secondary 47D07, 60J60, 35Q84, 15A24 |
| url | https://arxiv.org/abs/2605.25715 |