Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Some asymptotic properties of random walks on homogeneous spaces   Let $G$ be a connected semisimple real Lie group with finite center, and
$\mu$ a probability measure on $G$ whose support generates a Zariski-dense
subgroup of $G$. We consider the right $\mu$-random walk on $G$ and show that
each random trajectory spends most of its time at bounded distance of a
well-chosen Weyl chamber. We infer that if $G$ has rank one, and $\mu$ has a
finite first moment, then for any discrete subgroup $\Lambda \subseteq G$, the
$\mu$-walk and the geodesic flow on $\Lambda \backslash G$ are either both
transient, or both recurrent and ergodic, thus extending a well known theorem
due to Hopf-Tsuji-Sullivan-Kaimanovich dealing with the Brownian motion.

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The categories of this paper are: nucl-th
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The categories of this paper are: chem-ph
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The categories of this paper are: gr-qc
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The categories of this paper are: cmp-lg
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The categories of this paper are: ao-sci
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The categories of this paper are: quant-ph
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The categories of this paper are: q-bio
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The categories of this paper are: bayes-an
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The categories of this paper are: cs
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The categories of this paper are: cond-mat
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The categories of this paper are: math-ph
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The categories of this paper are: alg-geom
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The categories of this paper are: hep-ex
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The categories of this paper are: q-alg
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The categories of this paper are: patt-sol
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The categories of this paper are: nucl-ex