发表机构
University of Geneva; Institute of Mathematics of the National Academy of Sciences of Ukraine（日内瓦大学; 乌克兰国家科学院数学研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文建立正则一维扩散弱收敛的一般条件，基于Itô–McKean表示将随机收敛化为确定性问题，并应用于半透膜介质均匀化，得到反射、偏斜、黏性及断裂等极限过程。
AI 中文摘要
我们建立了正则一维扩散过程弱收敛的一般条件，允许其尺度函数、速度测度、状态空间及边界条件发生显著变化。我们的方法基于经典Itô–McKean表示，即通过单个布朗运动的时间变换和尺度变换来刻画正则扩散。这将许多随机收敛问题简化为关于积分、单调函数、复合函数及广义逆收敛的确定性问答题。作为主要应用，我们研究了具有紧密间隔半透膜的介质中扩散的各种均匀化机制，所得极限过程不仅包括反射和偏斜布朗运动，还包括偏斜-黏性布朗运动和断裂扩散。这给出了断裂扩散作为单个布朗运动的函数的表示，而无需引入独立的指数随机变量辅助序列。
英文摘要
We establish general conditions for the weak convergence of regular one-dimensional diffusion processes when their scale functions, speed measures, state spaces and boundary conditions may substantially vary. Our approach is based on the classical Itô--McKean representation of regular diffusions via time changes and scale transformations of a single Brownian motion. This reduces many stochastic convergence problems to deterministic questions concerning the convergence of integrals, monotone functions, compositions, and generalized inverses. As a main application, we investigate various homogenization regimes for diffusions in media with closely spaced semipermeable membranes, where the resulting limiting processes include not only reflected and skew Brownian motions, but also skew-sticky Brownian motions and snapping-out diffusions. This yields a representation of a snapping-out diffusion as a function of a single Brownian motion, without introducing an auxiliary sequence of independent exponential random variables.
Comments50 pages
