论文标题
反射的随机步行和不稳定的马丁边界
Reflected random walks and unstable Martin boundary
论文作者
论文摘要
我们介绍了一个在正象限中的二维反射随机步行的家庭，并研究了他们的马丁边界。虽然最小边界在系统上等于两个点的结合，但整个马丁边界在下面的意义上表现出不稳定的现象：如果与该模型相关的某些参数是合理的（分别为\非理性的），则马丁边界是可数的，而同型，$ \ \ \ \ \ \ \ \ m i \ mathbb z \ cup \ cup \ {\ \ {\ \ {\ fy fty \ homest，composh to to to to to to to to to to to to to to to to to to to to to to to to pm \\ ^ unc。 $ \ mathbb r \ cup \ {\ pm \ infty \} $）。这种不稳定现象在文献中非常罕见。在证明这一结果的过程中，我们获得了反射的随机步行的绿色函数的几个精确估计值，沿边界轴和任意大量的不均匀性域具有逃生概率。我们的方法将概率技术和一种随机步行的分析方法混合在一起，并在尺寸二的情况下进行大跳跃。
We introduce a family of two-dimensional reflected random walks in the positive quadrant and study their Martin boundary. While the minimal boundary is systematically equal to a union of two points, the full Martin boundary exhibits an instability phenomenon, in the following sense: if some parameter associated to the model is rational (resp.\ non-rational), then the Martin boundary is countable, homeomorphic to $\mathbb Z\cup\{\pm\infty\}$ (resp.\ uncountable, homeomorphic to $\mathbb R\cup\{\pm\infty\}$). Such instability phenomena are very rare in the literature. Along the way of proving this result, we obtain several precise estimates for the Green functions of reflected random walks with escape probabilities along the boundary axes and an arbitrarily large number of inhomogeneity domains. Our methods mix probabilistic techniques and an analytic approach for random walks with large jumps in dimension two.
