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  "body": "\n## Definition\n\nLet $G = (V, E)$ be a connected infinite [[graph|00C0BR7D7H]] in which every vertex has finitely many neighbours, with symmetric positive edge weights $c$.\nThe graph $G$ has the *Liouville property* if every bounded function $f \\colon V \\to \\R$ that is [[harmonic|WRY0M7SW59]] on all of $V$ is constant.\nIt has the *strong Liouville property* if every positive function that is harmonic on all of $V$ is constant.\nWithout weights, the edge weights are taken equal to $1$, which corresponds to the simple [[random walk|04VZE3FZXF]].\n\n## Remarks\n\nExample: $\\Z^d$ with unit weights has the Liouville property for every $d \\ge 1$.\nFor $d = 1$ this is direct, since the harmonic functions on $\\Z$ are $f(n) = an + b$, and such a function is bounded only when $a = 0$.\nNon-example: the $3$-regular [[tree|062D9KXJB1]] $T_3$ fails it.\nFix an edge $\\set{o, o'}$, and let $h(x)$ be the probability that the simple random walk from $x$ eventually stays in the component of $T_3$ minus this edge that contains $o'$.\nThe walk on $T_3$ is transient, so it crosses the edge finitely often, and $h$ is harmonic by conditioning on the first step.\nIts values lie in $[0, 1]$, and it is not constant.\nIndeed $h(x)$ tends to $1$ as $x$ moves away from the edge on the side of $o'$, and to $0$ on the side of $o$.\nThe strong Liouville property implies the Liouville property, since a bounded harmonic $f$ gives the positive harmonic function $f - \\inf f + 1$.\n",
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