On the inclusion of bounded harmonic functions of random walks

We investigate the conditions under which the space of bounded harmonic functions of a probability measure $\mu$ on a group $G$ is contained in that of another measure $\theta$. We establish that asymptotic commutativity, defined by the condition $\|\mu^{*t}*\theta - \theta*\mu^{*t}\|_{TV} \to 0$ as $t \to \infty$, is sufficient to guarantee the inclusion $H^\infty(G, \mu) \subseteq H^\infty(G, \theta)$, provided $\theta$ is absolutely continuous with respect to a convex combination of convolution powers of $\mu$. By employing martingale convergence techniques rather than ergodic-theoretic arguments, we demonstrate that this result holds without topological assumptions on $G$ (such as local compactness) and extends to general Markov chains. Furthermore, utilizing hitting models for the Poisson boundary, we characterise the inclusion $H^\infty(G, \mu) \subseteq H^\infty(G, \theta)$ as equivalent to the asymptotic invariance of $\theta$ under $\mu$ in the weak* topology. We apply these results to provide a probabilistic proof of the Choquet-Deny theorem for nilpotent groups, among other applications.