Linear-growth harmonic functions for nonsymmetric random walks on groups of polynomial growth

Abstract

Let $G$ be a finitely generated group of polynomial growth and let $\mu$ be an adapted, Abelian-centered probability measure with a finite exponential moment (not necessarily symmetric or finitely supported). We prove that every $\mu$-harmonic function of at most linear growth is globally Lipschitz. In particular, if $G$ is nilpotent, these functions are exactly the affine characters: $${\mathrm{HF}}_1(G,\mu )=\mathrm{LHF}(G,\mu )=P^1(G).$$ The analytic machinery required for this is a convolution-gradient estimate that is uniform over families of Abelian-centered, finitely supported probability measures satisfying fixed ellipticity and exponential-moment bounds. A quantitative induction-restriction theorem transfers $\mathrm{LHF}$ across finite-index subgroups. We also determine the normed structure of these spaces. In particular, on a nilpotent group, the Lipschitz seminorm of an affine character is exactly the dual stable norm of the word metric. Under the same nonsymmetric hypotheses, linear-growth harmonic functions modulo constants identify canonically with the virtual first cohomology. This identification is an isometry for the asymptotic Lipschitz seminorm. For the ordinary Lipschitz seminorm it is a contraction with bounded inverse; the inverse bound seems to depend on $\mu$, as shown by an infinite dihedral group example.

arXiv comment44 pages. The original submission arXiv:2512.06753 has now been split into two parts. This is the first part and the second part will be uploaded as a separate article later. Comments highly appreciated!