关于平面扩散在不连续分层介质中格林函数和马丁边界结构的研究
On the Green's functions and Martin boundary structure of a planar diffusion in a discontinuous layered medium
AI总结：
研究了平面扩散在不连续分层介质中的格林函数和马丁边界结构，推导了格林函数的拉普拉斯变换并确定了马丁边界。
摘要：
我们考虑在一个二维平面中，由上半平面和下半平面中不同的协方差矩阵所支配的二维扩散过程，且由两个指向x轴外的漂移向量所驱动。我们首先分析生成元以散度形式存在的情况，即当流体在界面处连续时的情况。然后我们将研究扩展到更广泛的类别，其在界面处的行为形成斜二维对称布朗运动的类比。我们对这个瞬态过程进行了详细的理论分析。我们的主要结果如下：(i) 我们推导了格林函数的显式拉普拉斯变换；(ii) 我们计算了格林函数在平面所有可能轨迹上的精确渐进行为；(iii) 我们确定了所有正调和函数，识别了完整的和最小的马丁边界，结果发现它们是不同的。马丁边界的非最小性是对于具有规则系数的扩散过程的一个值得注意的现象。为了获得该过程的分析描述，我们完全发展了所谓的核方法的三变量版本，通过推导并利用涉及未知格林函数拉普拉斯变换和两个已知核γ_+(x,y)和γ_{-}(x,z)的功能方程。引入独立的辅助变量y和z，与每个半平面相关联，是关键的想法。
英文摘要：
We consider a two-dimensional diffusion process in a two-layered plane, governed by distinct covariance matrices in the upper and lower half-planes and by two drift vectors pointed away from the $x$-axis. We first analyze the case where the generator of the process is in divergence form, that is, when the flux is continuous across the interface. Then we extend the study to a broader class of processes whose behavior at the interface forms an oblique two-dimensional analogue of the skew Brownian motion. We provide a detailed theoretical analysis of this transient process. Our main results are as follows: (i) we derive explicit Laplace transforms of the Green's functions; (ii) we compute exact asymptotics of the Green's functions along all possible trajectories in the plane; (iii) We determine all positive harmonic functions, identifying the full and minimal Martin boundaries, which turn out to be distinct. The nonminimality of the Martin boundary is a noteworthy phenomenon for diffusions with regular coefficients. To obtain an analytical description of the process, we fully develop a three-variable version of the so-called kernel method by deriving and exploiting a functional equation involving unknown Laplace transforms of Green's functions and two known kernels $γ_+(x,y)$ and $γ_{-}(x,z)$. The introduction of independent auxiliary variables $y$ and $z$, associated with each half-plane, is a key idea.
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