Limit Theorems for One-Dimensional Homogenized Diffusion Processes
Jaroslav I. Borodavka, Sebastian Krumscheid
TL;DR
The paper develops two limit theorems—a mean ergodic theorem and a central limit theorem—for a class of one-dimensional diffusions $X_\varepsilon$ with scale separation, in a coupled regime where the time horizon $T_\varepsilon$ diverges as $\varepsilon\to0$. Using transformed coordinates, invariant densities, occupation densities, and the Poisson equation, the authors derive an $\varepsilon$-dependent MET and CLT, alongside a convergence-in-probability result for $X_\varepsilon(T_\varepsilon)/\sqrt{T_\varepsilon}$. They then apply these results to a parameter estimation problem in homogenized models with perturbed data, proposing a minimum-distance estimator based on the invariant density’s characteristic function and proving consistency and asymptotic normality under the coupled limit. The findings provide a rigorous framework for asymptotic inference on homogenized diffusion parameters from multiscale observations, addressing gaps left by prior sequential limit results. Key techniques include localization, parabolic PDE Schauder estimates, occupation-density methods, and Poisson-equation analysis, all adapted to ε-dependent quantities.
Abstract
We present two limit theorems, a mean ergodic and a central limit theorem, for a specific class of one-dimensional diffusion processes that depend on a small-scale parameter $\varepsilon$ and converge weakly to a homogenized diffusion process in the limit $\varepsilon \rightarrow 0$. In these results, we allow for the time horizon to blow up such that $T_\varepsilon \rightarrow \infty$ as $\varepsilon \rightarrow 0$. The novelty of the results arises from the circumstance that many quantities are unbounded for $\varepsilon \rightarrow 0$, so that formerly established theory is not directly applicable here and a careful investigation of all relevant $\varepsilon$-dependent terms is required. As a mathematical application, we then use these limit theorems to prove asymptotic properties of a minimum distance estimator for parameters in a homogenized diffusion equation.