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.



Connection between degree of growth and return probabilities of random walks on Lie groups
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](http://www.normalesup.org/~tessera/Sobolev.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 $\simeq n^{-d/2}$.


