 Normalized stationary fluctuation-dissipation relation for a reaction network

Let $A$ be the drift Jacobian at a stable mean-field equilibrium, $S=\mathrm{diag}({\mu }_i)$, and $B={\sum }_r{a}_r{\nu }_r{\nu }_r^{\mathsf{T}}$ the reaction noise matrix. For normalized fluctuations, put $M=-S^{-1}AS$, $D=S^{-1}B{S}^{-1}$, and ${\eta }_{ij}=\mathrm{Cov}(X_i,X_j)/({\mu }_i{\mu }_j)$. 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