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Local limit theorems and equidistribution of random walks on the Heisenberg group. (English) Zbl 1083.60008
Let \(\mu\) be a compactly supported probability measure on the Heisenberg group \(G\). It is assumed that \(\mu\) is centered, that is \(\int p(x)\,d\mu (x)=0\) where \(p: G\to G/[G,G]\) is the canonical map, and that \(\mu (xH)<1\) for any proper closed subgroup \(H\) in \(G\) and any \(x\in G\). The author studies the limit behavior of the measures \(n^2\mu^n\) where \(\mu^n\) is the convolution power. Comparing \(\mu^n\) to the associated heat kernel, he gives uniform estimates for translates of a bounded set. This leads to an equidistribution result for the situation where \(G\) is a connected Lie group, \(\Gamma\) is a lattice in \(G\), and \(H\) is a closed subgroup of \(G\) consisting of unipotent elements and isomorphic to the Heisenberg group. For a measure \(\mu\) on \(H\) satisfying the above conditions, \(x\in G/\Gamma\), and \(f\in C_b(G/\Gamma )\), \[ \lim\limits_{n\to \infty }\int\limits_Hf(hx)\,d\mu^n(h)=\int\limits_Gf(g)\,dm_x(g) \] where \(m_x\) is the unique \(H\)-invariant ergodic probability measure on \(G/\Gamma\) whose support is the closure of the orbit \(Hx\).

60B15 Probability measures on groups or semigroups, Fourier transforms, factorization
43A05 Measures on groups and semigroups, etc.
43A85 Harmonic analysis on homogeneous spaces
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