= Normalized stationary fluctuation-dissipation relation for a reaction network
{title2=$M\eta+\eta M^{\mathsf T}=D$}
Assume all equilibrium <mean>[means] are strictly positive. Let $A$ be the drift <Jacobian matrix>[Jacobian] at a stable mean-field equilibrium, $S=\operatorname{diag}(\mu_i)$, and $B=\sum_ra_r\nu_r\nu_r^{\mathsf T}$ the reaction noise matrix. For normalized fluctuations, put $M=-S^{-1}AS$, $D=S^{-1}BS^{-1}$, and $\eta_{ij}=\operatorname{Cov}(X_i,X_j)/(\mu_i\mu_j)$. <stationary distribution>[Stationarity] of the linear-noise <covariance> gives the displayed <Continuous Lyapunov equation>. It applies to driven reaction networks without requiring thermal equilibrium. A mode with zero restoring eigenvalue and nonzero noise has no stationary <covariance>.
// Target: mathematical-biology.bigb
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