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arXiv · 2304.14551

Local limit theorems for random walks on nilpotent Lie groups

Timothée Bénard·Emmanuel Breuillard
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Abstract

We establish the (non-lattice) local limit theorem for products of i.i.d. random variables on an arbitrary simply connected nilpotent Lie group $G$, where the variables are allowed to be non-centered. Our result also improves on the known centered case by proving uniformity for two-sided moderate deviations and allowing measures with a moment of order $2(\dim G)^2$ without further regularity assumptions. As applications we establish a Ratner-type equidistribution theorem for unipotent walks on homogeneous spaces and obtain a new proof of the Choquet-Deny property in our setting.

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Timothée Bénard, Emmanuel Breuillard. 2023-04-27. Local limit theorems for random walks on nilpotent Lie groups. https://arxiv.org/abs/2304.14551

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