<p>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 <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\varepsilon $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ε</mi> </math></EquationSource> </InlineEquation> and converge weakly to a homogenized diffusion process in the limit <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\varepsilon \rightarrow 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ε</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>. In these results, we allow for the time horizon to blow up such that <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$T_\varepsilon \rightarrow \infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>T</mi> <mi>ε</mi> </msub> <mo stretchy="false">→</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation> as <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\varepsilon \rightarrow 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ε</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>. The novelty of the results arises from the circumstance that many quantities are unbounded for <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\varepsilon \rightarrow 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ε</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, so that formerly established theory is not directly applicable here and a careful investigation of all relevant <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\varepsilon $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ε</mi> </math></EquationSource> </InlineEquation>-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.</p>

Limit Theorems for One-Dimensional Homogenized Diffusion Processes

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2026年3月2日

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Jaroslav I. Borodavka,
Sebastian Krumscheid

摘要

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 $ $ε$ and converge weakly to a homogenized diffusion process in the limit $\varepsilon \rightarrow 0$ $ε \to 0$ . In these results, we allow for the time horizon to blow up such that $T_\varepsilon \rightarrow \infty $ $T_{ε} \to \infty$ as $\varepsilon \rightarrow 0$ $ε \to 0$ . The novelty of the results arises from the circumstance that many quantities are unbounded for $\varepsilon \rightarrow 0$ $ε \to 0$ , so that formerly established theory is not directly applicable here and a careful investigation of all relevant $\varepsilon $ $ε$ -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.