For closed manifolds, we know that the Poisson boundary is trivial due to compactness and for radially symmetric manifolds for which diffusion is one dimensional, there are A Brief Introduction to Brownian motion on a Riemannian Manifold by Elton P Hsu and Kinetic Brownian motion on Riemannian manifolds by Angst, Ballieul and Tardif which made me wonder if the only Riemannian manifolds which admit one-dimensional diffusions are the radially symmetric ones. Furthermore, what do we know about Poisson boundaries in the general setting. I first asked this question in maths stack exchange 17 days ago but I did not get any answer there.
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