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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 \(Hx\).

MSC:
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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