We study harmonic functions with respect to the Riemannian metric

in the upper half space \(\mathbb {R}_{+}^{n}=\{\left( x_{1},\ldots ,x_{n}\right) \in \mathbb {R}^{n}:x_{n}>0\}\). They are called \(\alpha \)-hyperbolic harmonic. An important result is that a function f is \(\alpha \)-hyperbolic harmonic íf and only if the function \(g\left( x\right) =x_{n}^{-\frac{ 2-n+\alpha }{2}}f\left( x\right) \) is the eigenfunction of the hyperbolic Laplace operator \(\bigtriangleup _{h}=x_{n}^{2}\triangle -\left( n-2\right) x_{n}\frac{\partial }{\partial x_{n}}\) corresponding to the eigenvalue \(\ \frac{1}{4}\left( \left( \alpha +1\right) ^{2}-\left( n-1\right) ^{2}\right) =0\). This means that in case \(\alpha =n-2\), the \(n-2\)-hyperbolic harmonic functions are harmonic with respect to the hyperbolic metric of the Poincaré upper half-space. We are presenting some connections of \(\alpha \)-hyperbolic functions to the generalized hyperbolic Brownian motion. These results are similar as in case of harmonic functions with respect to usual Laplace and Brownian motion.

我们研究关于黎曼度量的谐波函数

在上半空间\（\ mathbb {R} _ {+} ^ {n} = \ {\ left（x_ {1}，\ ldots，x_ {n} \ right）\ in \ mathbb {R} ^ { n}：x_ {n}> 0 \} \）。它们称为\（\ alpha \）-双曲谐波。重要的结果是，函数f是\（\ alpha \）-双曲调和，仅当函数\（g \ left（x \ right）= x_ {n} ^ {-\ frac {2-n + \ alpha } {2}} f \ left（x \ right）\）是双曲Laplace算子\（\ bigtriangleup _ {h} = x_ {n} ^ {2} \ triangle-\ left（n-2 \ right）x_ {n} \ frac {\ partial} {\ partial x_ {n}} \）对应于特征值\ {\ frac {1} {4} \ left（\ left（\ alpha（\ alpha +1 \ right）） ^ {2}-\ left（n-1 \ right）^ {2} \ right）= 0 \）。这意味着\（\ alpha = n-2 \），\（n-2 \）-双曲调和函数是相对于Poincaré上半空间的双曲度量的调和。我们正在介绍\（\ alpha \）-双曲函数与广义双曲布朗运动的一些联系。这些结果与关于通常的拉普拉斯运动和布朗运动的谐波函数的情况相似。