Bounded Harmonic Functions on Products with a Parabolic Factor

Abstract

We prove that if $M$ is a connected complete parabolic Riemannian manifold and $N$ is a connected complete stochastically complete Riemannian manifold, then every bounded harmonic function on $M\times N$ is independent of the $M$-variable. Equivalently, pullback by the second projection induces an isometric isomorphism from the space of bounded harmonic functions on $N$ onto that on $M\times N$. In particular, the product of two parabolic manifolds has the bounded Liouville property, thereby answering Problem~16 of Grigor'yan's survey in the affirmative. The key analytic input is a total-variation memory-loss property of the heat kernel on a parabolic manifold. We establish this property by showing that the time-one heat kernel defines an aperiodic Harris recurrent transition kernel and then applying the row-merging theorem of Jamison and Orey.