# zbMATH — the first resource for mathematics

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
##### Keywords:
convolution power
Full Text: