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.
| Comments: | Long version with all details |
| Subjects: | Probability (math.PR) |
| MSC classes: | Primary 60J35, Secondary 47D07, 60J60, 35Q84, 15A24 |
| Cite as: | arXiv:2605.25715 [math.PR] |
| (or arXiv:2605.25715v1 [math.PR] for this version) | |
| https://doi.org/10.48550/arXiv.2605.25715 arXiv-issued DOI via DataCite (pending registration) |
From: Gerald Trutnau [view email]
[v1]
Mon, 25 May 2026 11:16:37 UTC (40 KB)
此内容由惯性聚合(RSS阅读器)自动聚合整理，仅供阅读参考。 原文来自 — 版权归原作者所有。