论文标题
共形Skorokhod嵌入和相关的极端问题
Conformal Skorokhod embeddings and related extremal problems
论文作者
论文摘要
Skorokhod嵌入问题（CSEP）是经典问题的平面变体，该解决方案现在是一个简单连接的域$ d \ subset \ subset \ mathbb {c} $，其退出时间嵌入了给定的概率分布$μ$，通过将停止的Brownian运动投射到真实的轴心上。在本文中，我们通过证明解决方案域的主要迪里奇特征值的一般界限来探讨CSEP的两个新研究方向，并通过相应的$μ$以及提出相关的极端问题。此外，我们给出了一个极端域$ \ mathbb {u} $的新的非平凡示例，该示例在所有域中都达到了所有域，该范围在$ [-1,1] $上求解CSEP的所有域上都可以解决CSEP。值得注意的是，$ \ mathbb {u} $的边界与平面曲线缩短流的严格收割机转换解决方案有关。尖锐下限证明的新工具是简单连接的平面域的正交投影的宽度与我们在论文中开发的谐波测量的支持之间的精确关系。上限依赖于近来出现在文献中的扭转函数的光谱界限。
The conformal Skorokhod embedding problem (CSEP) is a planar variant of the classical problem where the solution is now a simply connected domain $D\subset\mathbb{C}$ whose exit time embeds a given probability distribution $μ$ by projecting the stopped Brownian motion onto the real axis. In this paper we explore two new research directions for the CSEP by proving general bounds on the principal Dirichlet eigenvalue of a solution domain in terms of the corresponding $μ$ and by proposing related extremal problems. Moreover, we give a new and nontrivial example of an extremal domain $\mathbb{U}$ that attains the lowest possible principal Dirichlet eigenvalue over all domains solving the CSEP for the uniform distribution on $[-1,1]$. Remarkably, the boundary of $\mathbb{U}$ is related to the Grim Reaper translating solution to the curve shortening flow in the plane. The novel tool used in the proof of the sharp lower bound is a precise relationship between the widths of the orthogonal projections of a simply connected planar domain and the support of its harmonic measure that we develop in the paper. The upper bound relies on spectral bounds for the torsion function which have recently appeared in the literature.
