Consider an irreducible continuous-time Markov chain on $\{1,\dots,n\}$ with generator $Q$ and stationary distribution $\pi$. Pass to the Fisher-conjugated generator $L_\pi = D_\pi^{1/2} Q D_\pi^{-1/2}$ and restrict to the $(n{-}1)$-dimensional tangent space at $\pi$ (the subspace orthogonal to $\sqrt{\pi}$). In these reduced Euclidean Fisher coordinates, write the restricted operator as $\widehat{K} = \widehat{G} + \widehat{J}$ (symmetric + skew-symmetric parts).
The Donsker-Varadhan rate function for the empirical measure has a Hessian $H_{DV}$ at $\pi$ in these coordinates. The detailed-balance reference has Hessian $H_0 = -\frac{1}{2}\widehat{G}$.
A completing the square arguement in the DV variational envelope gives:
$$H_{DV} - H_0 \;=\; \tfrac{1}{4}\;\widehat{J}\;H_0^{-1}\;\widehat{J}^{\,T}$$
That is, the nonequilibrium correction to the Donsker-Varadhan Hessian factors exactly as a Gram operator, the skew channel, weighted by the inverse detailed-balance backbone, wrapped into a positive semidefinite square.
The proof is short.
Setting $S = -\widehat{G}$, the second-order DV envelope in reduced coordinates is $\mathcal{L}(x,y) = -y^T S y + x^T(S+\widehat{J})y$.
Completing the square in $y$ and supremising gives $I_{DV}(\mu(x)) = \frac{1}{4}x^T(S + \widehat{J})^T S^{-1}(S+\widehat{J})x + O(|x|^3)$.
Since $\widehat{J}^T = -\widehat{J}$, the cross terms simplify: $(S-\widehat{J})S^{-1}(S+\widehat{J}) = S + \widehat{J}S^{-1}\widehat{J}^T$. Substituting $H_0 = \frac{1}{2}S$ gives the result.
This factorisation implies that $\operatorname{rank}(\Delta_{DV}) = \operatorname{rank}(\widehat{J})$, that detailed balance is equivalent to $\Delta_{DV} = 0$, and that finite-rank observation of the nonequilibrium correction reduces to a Ky Fan / Eckart-Young problem on the singular values of $\widehat{J}H_0^{-1/2}$.
All the ingredients, the DV variational formula (1975), Fisher conjugation, the symmetric-skew split in the stochastic thermodynamics literature (e.g. Maes-Netočný-Wynants 2008), have been available for decades. Yet I have not found this specific factorisation in the large deviations, information geometry, or stochastic thermodynamics literature, despite a careful search.
Has this identity, or an equivalent formulation, appeared before?
Ref: J. R. Dunkley, Finite Observation, preprint (2025), doi:10.5281/zenodo.19102609.
