On the inclusion of bounded harmonic functions of random walks
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866917223611760640 |
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| author | Hartman, Yair Hrušková, Aranka Segev, Omer |
| author_facet | Hartman, Yair Hrušková, Aranka Segev, Omer |
| contents | We investigate the conditions under which the space of bounded harmonic functions of a probability measure $μ$ on a group $G$ is contained in that of another measure $θ$. We establish that asymptotic commutativity, defined by the condition $\|μ^{*t}*θ- θ*μ^{*t}\|_{TV} \to 0$ as $t \to \infty$, is sufficient to guarantee the inclusion $H^\infty(G, μ) \subseteq H^\infty(G, θ)$, provided $θ$ is absolutely continuous with respect to a convex combination of convolution powers of $μ$. 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 $H^\infty(G, μ) \subseteq H^\infty(G, θ)$ as equivalent to the asymptotic invariance of $θ$ under $μ$ in the weak* topology. We apply these results to provide a probabilistic proof of the Choquet-Deny theorem for nilpotent groups, among other applications. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_18304 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the inclusion of bounded harmonic functions of random walks Hartman, Yair Hrušková, Aranka Segev, Omer Probability Dynamical Systems Group Theory 60G50 (Primary) 20P05, 60J10 (Secondary) We investigate the conditions under which the space of bounded harmonic functions of a probability measure $μ$ on a group $G$ is contained in that of another measure $θ$. We establish that asymptotic commutativity, defined by the condition $\|μ^{*t}*θ- θ*μ^{*t}\|_{TV} \to 0$ as $t \to \infty$, is sufficient to guarantee the inclusion $H^\infty(G, μ) \subseteq H^\infty(G, θ)$, provided $θ$ is absolutely continuous with respect to a convex combination of convolution powers of $μ$. 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 $H^\infty(G, μ) \subseteq H^\infty(G, θ)$ as equivalent to the asymptotic invariance of $θ$ under $μ$ in the weak* topology. We apply these results to provide a probabilistic proof of the Choquet-Deny theorem for nilpotent groups, among other applications. |
| title | On the inclusion of bounded harmonic functions of random walks |
| topic | Probability Dynamical Systems Group Theory 60G50 (Primary) 20P05, 60J10 (Secondary) |
| url | https://arxiv.org/abs/2601.18304 |