Take two graphs (of bounded valency) or manifolds (f bounded geometry) $G$ and $G'$. Assume there is a quasi-isometry $f:G \to G'$, and assume the Poisson or Martin boundary of $G$ is known, what may one say about the Poisson or Martin boundary of $G'$ (or how does $f$ transforms Poisson or Martin boundaries)?

Take two graphs (of bounded valency) or manifolds (f bounded geometry) $G$ and $G'$. Assume there is a quasi-isometry $f:G \to G'$, and assume the Poisson or Martin boundary of $G$ is known, what may one say about the Poisson or Martin boundary of $G'$ (or how does $f$ transforms Poisson or Martin boundaries)?

EDIT: a map $f:X \to Y$ between two metric spaces $(X,d_X)$ and $(Y,d_Y)$ is a quasi-isometry if $\exists C \geq 0$ and $D \geq 1$ such that

• $\forall x,x' \in X$, $D^{-1} d_Y \big( f(x), f(x') \big) - C \leq d_X(x,x') \leq D d_Y \big( f(x), f(x') \big) + C $

• $\forall y \in Y, d_Y \big(f(X),y \big) < C$