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

TL;DR

This paper analyzes the white-noise limit of the colored linear inverse model (LIM), showing that as the correlation time approaches zero, the colored LIM converges to the classical LIM, clarifying the behavior of estimation formulas.

Contribution

It provides a stochastic differential equations perspective on the white-noise limit of the colored LIM, reconciling convergence of dynamics with estimation challenges.

Findings

Colored LIM reduces to classical LIM as tau -> 0

Stationary covariance satisfies fluctuation-dissipation relation in the limit

Numerical illustrations confirm convergence behavior

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.