论文标题
反映曲线楔的随机步行
Reflecting random walks in curvilinear wedges
论文作者
论文摘要
我们在一个无限的平面域中研究一个随机步行（马尔可夫链），其边界由形式的两个曲线描述为$ x_2 = a^+ x_1^{β^+} $和$ x_2 = -a^ - x_1^ - x_1^{β^ - } $，带有$ x_1 \ geq 0 $。在域的内部，随机步行的漂移为零和给定的增量协方差矩阵。从边界的上部和下部的附近，步行以给定的角度$α^+$或$α^ - $向内部移回了内部，到相关的内向向内正常向量。在这里，我们关注$α^+$和$α^ - $相等但相反的情况，其中包括正常反射的情况。对于$ 0 \ leqβ^+，β^ - <1 $，我们根据模型参数确定复发和瞬态之间的相变，并通过通道时间矩量化复发。
We study a random walk (Markov chain) in an unbounded planar domain whose boundary is described by two curves of the form $x_2 = a^+ x_1^{β^+}$ and $x_2 = -a^- x_1^{β^-}$, with $x_1 \geq 0$. In the interior of the domain, the random walk has zero drift and a given increment covariance matrix. From the vicinity of the upper and lower sections of the boundary, the walk drifts back into the interior at a given angle $α^+$ or $α^-$ to the relevant inwards-pointing normal vector. Here we focus on the case where $α^+$ and $α^-$ are equal but opposite, which includes the case of normal reflection. For $0 \leq β^+, β^- < 1$, we identify the phase transition between recurrence and transience, depending on the model parameters, and quantify recurrence via moments of passage times.
