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<p>In this paper, we study the long-time behavior of time-inhomogeneous diffusion processes in one dimension. Utilizing the probabilistic coupling method, we introduce a sufficient condition, inspired by Chen and Wang (Trans Am Math Soc, 349(3):1239–1267, 1997), to analyze the convergence of the transition kernels starting from different initial locations under a specific Wasserstein metric. Subsequently, we apply the main results to derive quantitative convergence rates for time-inhomogeneous diffusion processes by establishing rigorous conditions on the drift and diffusion coefficients. Our framework encompasses classical examples characterized by logarithmic, algebraic, and stretched exponential convergence rates. In particular, we offer a new perspective for evaluating the non-exponential convergence of time-homogeneous diffusions.</p>

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Long-Time Behavior of Time-Inhomogeneous Diffusion Processes Under the Wasserstein Distance

  • 期刊论文

  • 发表日期：

    2026年3月2日

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  • Xiaobin Sun,
  • Yingchao Xie,
  • Dong Yao

摘要

In this paper, we study the long-time behavior of time-inhomogeneous diffusion processes in one dimension. Utilizing the probabilistic coupling method, we introduce a sufficient condition, inspired by Chen and Wang (Trans Am Math Soc, 349(3):1239–1267, 1997), to analyze the convergence of the transition kernels starting from different initial locations under a specific Wasserstein metric. Subsequently, we apply the main results to derive quantitative convergence rates for time-inhomogeneous diffusion processes by establishing rigorous conditions on the drift and diffusion coefficients. Our framework encompasses classical examples characterized by logarithmic, algebraic, and stretched exponential convergence rates. In particular, we offer a new perspective for evaluating the non-exponential convergence of time-homogeneous diffusions.