曲线楔中的反射随机游动
Reflecting random walks in curvilinear wedges
摘要：
研究了无界平面区域中的随机游动（马尔可夫链），该区域的边界由两条曲线$x_2 = a^+ x_1^{β^+}$和$x_2 = -a^- x_1^{β^-}$描述，其中$x_1 \geq 0$ .在区域的内部，随机游动具有零漂移和给定的增量协方差阵。从边界的上部和下部附近，行走以给定的角度$α^+$或$α^-$向相关的向内指向的法向量漂移回到内部。这里，我们关注$α^+$和$α^-$相等但相反的情况，这包括正常反射的情况。对于$0 \leq β^+, β^- < 1$，我们确定复发和瞬态之间的相变，根据模型参数，并通过时刻的通过时间量化复发。
英文摘要：
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.
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