I have started reading a paper of Colding and Minicozzi, where they prove that on a complete Riemannian manifold $M$ of non-negative Ricci curvature, the space of harmonic functions of growth order at most $d$ is finite dimensional. It is known that these $M$ have at most polynomial volume growth.

I am wondering how much the growth of $M$ is related to the dimensionality of harmonic functions. For example, is it known what happens when the volume growth is known to be not polynomial? To be more specific, suppose we ask the analogous question on the hyperbolic space: are the harmonic functions of polynomial growth on $\mathbb{H}^n$ infinite dimensional? This is mainly a reference request.