Energy balancing integrals, orthogonal polynomial systems, and matrix Lyapunov equations

Starting from a steady-state energy–balance law for noisy, linearly damped fields, we derive an operator covariance equation in Lyapunov–Sylvester form. On the unit disk with self-adjoint boundary conditions, the corresponding generator is diagonalized by the Zernike polynomials, and finite-mode projection yields a matrix Lyapunov equation for modal covariances. We prove explicit a priori truncation-error bounds tailored to Zernike systems: in the diagonal (or diagonally dominant) case the operator-norm tail admits a closed-form expression and, for Kolmogorov-type spectra, decays at a rate $ \mathcal{O}(N^{-7/3}) $; for general Hilbert–Schmidt noise covariances we obtain Hilbert–Schmidt tail bounds with explicit dependence on system parameters and dissipation rates. We further extend the formulation to exponentially correlated (Ornstein-Uhlenbeck) forcing via an augmented-state Lyapunov/Sylvester construction, yielding closed-form denominator shifts and a covariance-based inversion for the OU correlation time $ \tau $. Numerical examples validate the Lyapunov solves, the derived error bounds, and the $ \tau $ recovery procedure.