论文标题
$ {\ mathbb {z}}^d $的椭圆harnack不平等
Elliptic Harnack Inequality for ${\mathbb{Z}}^d$
论文作者
论文摘要
我们证明了在$ {\ Mathbb {z}}}^d $上的非阴性谐波函数的规模椭圆harnack不等式（EHI）。本说明的目的是在本科级别可访问的$ {\ mathbb {z}}^d $中提供简化的EHI的概率证明。我们使用局部中央限制定理用于$ {\ Mathbb {z}}^d $上的简单对称随机步行，以建立$ n $ step概率函数的高斯界限。然后，统一的绿色不平等和经典的Balayage公式表示EHI。
We prove the scale invariant Elliptic Harnack Inequality (EHI) for non-negative harmonic functions on ${\mathbb{Z}}^d$. The purpose of this note is to provide a simplified self-contained probabilistic proof of EHI in ${\mathbb{Z}}^d$ that is accessible at the undergraduate level. We use the Local Central Limit Theorem for simple symmetric random walks on ${\mathbb{Z}}^d$ to establish Gaussian bounds for the $n$-step probability function. The uniform Green inequality and the classical Balayage formula then imply the EHI.
