According to Score-Based Generative Modeling through Stochastic Differential Equations, we have the following.
Definition (Forward SDE) $$ \mathrm{d}\mathbf{x} = \mathbf{F}(\mathbf{x}, t)\mathrm{d}t + \mathbf{G}(\mathbf{x}, t)\mathrm{d}\mathbf{w}, $$ where $\mathbf{x} \in \mathbb{R}^n$ is the state, $t \in \mathbb{R}$ is the forward time, $\mathbf{F}: \mathbb{R}^n \to \mathbb{R}^n$ is the drift coefficient, $\mathbf{G}: \mathbb{R}^n \to \mathbb{R}^{n \times n}$ is the diffusion coefficient, and $\mathbf{w} \in \mathbb{R}^n$ is the Weiner process.
Theorem (Probability Flow ODE) $$ \frac{\mathrm{d}\mathbf{x}}{\mathrm{d}t} = \mathbf{F}(\mathbf{x}, t) - \frac{1}{2}\nabla_\mathbf{x} \cdot(\mathbf{G}(\mathbf{x}, t)\mathbf{G}(\mathbf{x}, t)^\top) - \frac{1}{2} \mathbf{G}(\mathbf{x}, t)\mathbf{G}(\mathbf{x}, t)^\top\nabla_\mathbf{x}\log(p(\mathbf{x}, t)), $$ where the marginal PDF $p(\mathbf{x}(t), t)$ is equivalent for solutions $\mathbf{x}(t)$ of the Forward SDE.
Lemma (Linear Time-Invariant SDE)
Let $\mathbf{F}(\mathbf{x},t) = \mathbf{A}\mathbf{x}$ with $\mathbf{A} = \mathbf{A}^\top$ and $\mathbf{G}(\mathbf{x}, t) = \mathbf{B}$, yielding a linear time-invariant Forward SDE: $$ \mathrm{d}\mathbf{x} = \mathbf{A}\mathbf{x}\mathrm{d}t + \mathbf{B}\mathrm{d}\mathbf{w}. $$ and a Probability Flow ODE given by $$ \frac{\mathrm{d}\mathbf{x}}{\mathrm{d}t} = \mathbf{A}\mathbf{x} - \frac{1}{2}\mathbf{B}\mathbf{B}^\top\nabla_\mathbf{x}\log(p(\mathbf{x})) = \mathbf{C}\mathbf{x}. $$
Question
Assume that there exists a stationary distribution of the Linear Time-Invariant SDE given by $$ p_\infty = \mathcal{N}(\mathbf{\mu}_\infty, \mathbf{\Sigma}_\infty). $$ Then, how does the stability of the Linear Time-Invariant Probability Flow ODE (i.e. the eigenvalues of $\mathbf{C}$) relate to this stationary distribution?