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  "body": "\n## Definition\n\nLet $\\Gamma$ be a countable group and let $\\mu$ be a [[probability measure|059TJN0NZ1]] on $\\Gamma$.\nIt defines the [[random walk|WAS9EANSJJ]] $X_n = g_1 g_2 \\cdots g_n$ with independent steps $g_k$ of law $\\mu$.\nA function $f \\colon \\Gamma \\to \\R$ is $\\mu$-*harmonic* if $f(g) = \\sum_{h \\in \\Gamma} \\mu(h) f(gh)$ for every $g \\in \\Gamma$.\nLet $H^\\infty(\\Gamma, \\mu)$ be the space of bounded $\\mu$-harmonic functions.\nA *Poisson boundary* of $(\\Gamma, \\mu)$ is a standard probability space $(B, \\nu)$ with a measurable action of $\\Gamma$ such that:\n\n1. $\\nu$ is $\\mu$-stationary: $\\sum_{h \\in \\Gamma} \\mu(h)\\, h_* \\nu = \\nu$.\n2. The Poisson transform $\\varphi \\mapsto f_\\varphi$, with $f_\\varphi(g) = \\int_B \\varphi(g \\xi) \\, d\\nu(\\xi)$, is a bijection from $L^\\infty(B, \\nu)$ onto $H^\\infty(\\Gamma, \\mu)$.\n\n## Remarks\n\nClause 1 makes each $f_\\varphi$ harmonic: $\\sum_h \\mu(h) \\int_B \\varphi(gh\\xi) \\, d\\nu(\\xi) = \\int_B \\varphi(g\\xi) \\, d\\Big(\\sum_h \\mu(h) h_* \\nu\\Big)(\\xi) = f_\\varphi(g)$.\nThe Poisson boundary exists and is unique up to a $\\Gamma$-equivariant measure isomorphism.\nOne construction takes the space of ergodic components of the shift on the space of sample paths of the walk, with the image of the path measure.\nExample: let $\\Gamma = \\Z^d$ and let the support of $\\mu$ generate $\\Z^d$ as a group.\nBy the Choquet–Deny theorem the bounded harmonic functions are constant, so a one-point space is a Poisson boundary.\nNon-example: for the free group $F_2$ on $a, b$ and $\\mu$ uniform on $\\set{a, a^{-1}, b, b^{-1}}$, a one-point space fails clause 2, since there are nonconstant bounded harmonic functions.\nIn that case a Poisson boundary is the space of infinite reduced words in $a^{\\pm 1}, b^{\\pm 1}$ with the law of the limit of $X_n$.\nFor the simple random walk on a Cayley graph, a trivial Poisson boundary is the same as the [[Liouville property|Y1EGR8DF84]] of the graph.\n",
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