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Let $G$ be a finitely generated group of polynomial growth, let $\mu$ be a non-degenerate symmetric probability measure with finite support on $G$, and let $d$ be the degree of growth of $G$. Varopoulos proved that $\mu^{2n}(e) = O(n^{- \frac{d}{2}})$ and $n^{- \frac{d}{2}} = O(\mu^{2n}(e)) $ (we write $\mu^{2n}(e) \sim n^{- \frac{d}{2}}$), where $\mu^{2n}$ is the $2n$-th convolution power of $\mu$. Letting $p_{2n}$ be the probability that the random walk on $G$ associated with $\mu$ returns to the origin after $2n$ steps, Varopoulos' result implies that $p_{2n} \sim n^{- \frac{d}{2}}$ also.

Now let $G$ be a Lie group of polynomial growth. Does a similar result to Varopoulos' exist in this setting? References on this topic would be much appreciated.

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Every locally compact group of polynomial growth is QI to a simply connected nilpotent Lie group [Losert: On the structure of groups with polynomial growth, Math. Z. 195(1) (1987) 109-117], and probability of return is stable under quasi-isometries [Tessera Large scale Sobolev inequalities on metric measure spaces and applications. Rev. Mat. Iberoam. 24 (2008), no. 3, 825--864:: pdf]. Whence the reduction to simply connected nilpotent Lie groups due to Varopoulos (Varopoulos did not restrict to finitely generated nilpotent groups or nilpotent Lie groups with lattices). In conclusion, for every locally compact group of polynomially bounded growth, the growth is $\simeq n^d$ with $d$ integer and the probability of return is $\simeq n^{-d/2}$.

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  • $\begingroup$ I am reading the Tessera paper, but am having trouble understanding exactly which result there implies that probability of return is stable under quasi-isometries for Lie groups (I have seen this result for finitely generated groups). Is it his main result, Theorem 1 in the paper? I know that specifying the "degree of decay" of the isoperimetric profile is equivalent to doing so for the return probability, but that result discusses $L^p$-isoperimetric profiles, which don't seem to be the same thing. $\endgroup$
    – user47125
    Commented Feb 19, 2014 at 21:55
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    $\begingroup$ It's indeed a bit implicit. On page 3 he says "In particular, we obtain that upper bounds on probability of return of symmetric random walks are characterized by large-scale Sobolev inequalities and therefore are invariant under large-scale equivalence (see Theorem 3.5 for a precise statement)." The statement is precise in itself but does not refer explicitly to those facts about QI-invariance; hopefully this follows from a more careful reading. $\endgroup$
    – YCor
    Commented Feb 19, 2014 at 23:31

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