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On the White-Noise Limit of the Colored Linear Inverse Model

Cristian Martinez-Villalobos

Abstract

A recent paper by Lien et al. (2025) introduces the "colored linear inverse model" (colored LIM), in which stochastic forcing is modeled using Ornstein-Uhlenbeck colored noise rather than idealized white noise. In that work, it is shown that the derivative-based identification formulas used to estimate model parameters do not admit a regular white-noise limit due to the loss of differentiability of the lag-correlation function at zero lag. Here we revisit the white-noise limit from the perspective of the underlying stochastic differential equations. Treating the colored LIM as an augmented Ornstein-Uhlenbeck system, we show that as the correlation time tau -> 0 the colored-noise-driven system reduces to the classical LIM, and the corresponding stationary covariance satisfies the standard fluctuation-dissipation relation. Re-examining the same linear system used by Lien et al. (2025), we illustrate this convergence numerically. These results highlight a distinction between the singular behavior of derivative-based identification formulas and the regular limiting behavior of the underlying stochastic model. Taken together with recent results showing convergence of estimated parameters in the white-noise limit, they provide a consistent interpretation in which the colored LIM recovers the classical LIM at the level of stochastic dynamics even though certain estimation procedures become ill-defined in that limit.

On the White-Noise Limit of the Colored Linear Inverse Model

Abstract

A recent paper by Lien et al. (2025) introduces the "colored linear inverse model" (colored LIM), in which stochastic forcing is modeled using Ornstein-Uhlenbeck colored noise rather than idealized white noise. In that work, it is shown that the derivative-based identification formulas used to estimate model parameters do not admit a regular white-noise limit due to the loss of differentiability of the lag-correlation function at zero lag. Here we revisit the white-noise limit from the perspective of the underlying stochastic differential equations. Treating the colored LIM as an augmented Ornstein-Uhlenbeck system, we show that as the correlation time tau -> 0 the colored-noise-driven system reduces to the classical LIM, and the corresponding stationary covariance satisfies the standard fluctuation-dissipation relation. Re-examining the same linear system used by Lien et al. (2025), we illustrate this convergence numerically. These results highlight a distinction between the singular behavior of derivative-based identification formulas and the regular limiting behavior of the underlying stochastic model. Taken together with recent results showing convergence of estimated parameters in the white-noise limit, they provide a consistent interpretation in which the colored LIM recovers the classical LIM at the level of stochastic dynamics even though certain estimation procedures become ill-defined in that limit.

Paper Structure

This paper contains 5 sections, 17 equations, 1 figure.

Table of Contents

  1. Problem Setup
  2. Interpreting the Discrepancy
  3. Recovery of the Classical LIM at the Level of the Stochastic Dynamics
  4. Numerical demonstration using Lien et al. (2025) system
  5. Conclusions

Figures (1)

  • Figure 1: Convergence test using the system of lien_colored_2025, Eq. (32). The plot shows the relative Frobenius error $\|\mathbf{C}_{xx}(\tau)-\mathbf{C}_{\text{white}}\|_F / \|\mathbf{C}_{\text{white}}\|_F$ as a function of correlation time $\tau$ (in months) on logarithmic axes. The error decreases monotonically as $\tau\to 0$, demonstrating recovery of the classical LIM.