非光滑细管中带反射的Wiener过程
Wiener Process with Reflection in Non-Smooth Narrow Tubes
摘要：
本文研究了宽度ε<<1的窄管内绕x轴的瞬态反射Wiener过程。在下面的意义上，假设管是（渐近）非光滑的。令$V^ε(x)$为管的横截面的体积。我们假设当ε->0时，$V^ε(x)/ε$在适当的意义上收敛于非光滑函数。该极限函数可以由光滑函数、阶梯函数和狄拉克δ分布组成。在此假设下，我们证明了Wiener过程的x分量弱收敛于Markov过程，该Markov过程在远离不连续点的地方表现得像标准的扩散过程，并且必须满足不连续点处的某些粘合条件。
英文摘要：
Wiener process with instantaneous reflection in narrow tubes of width ε<<1 around axis x is considered in this paper. The tube is assumed to be (asymptotically) non-smooth in the following sense. Let $V^ε(x)$ be the volume of the cross-section of the tube. We assume that $V^ε(x)/ε$ converges in an appropriate sense to a non-smooth function as ε->0. This limiting function can be composed by smooth functions, step functions and also the Dirac delta distribution. Under this assumption we prove that the x-component of the Wiener process converges weakly to a Markov process that behaves like a standard diffusion process away from the points of discontinuity and has to satisfy certain gluing conditions at the points of discontinuity.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
