All Questions

Let $G$ be a finitely generated group and let $\mu$ be a symmetric non-degenerate measure on $G$. By strong Liouville property for $(G, \mu)$, we mean that every positive $\mu$-harmonic function on $G$...

We consider a harmonic function $u:\mathbb{R}^d \to \mathbb{R}$ $(\Delta u=0).$ Suppose that $$\exists \alpha,\beta \in \mathbb{R},\forall x\in \mathbb{R}^d,u(x)\leq \alpha |x|+\beta.$$ Therefore $u-u(...

It seems classically known (and mentioned in several papers without reference) that there exist bounded non-constant harmonic functions on the hyperbolic space $\mathbb{H}^n, n \geq 2$. I am ...

Given two Markov chains with respective state-spaces $S$ and $T$, say that a function $\phi$ from $S$ to $T$ is holomorphic iff for all states $t \in T$, every real-valued function $f$ on $T$ that is ...