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<p>We reduce the local limit theorem for a non-compact semisimple Lie group acting on its symmetric space to establishing that a natural operator associated to the measure is quasicompact. Under strong Diophantine assumptions on the underlying measure, we deduce the necessary spectral results for the operator in question. We thereby give the first examples of finitely supported measures satisfying such a local limit theorem. Moreover, quantitative error rates for the local limit theorem are proved under additional assumptions.</p>

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Local limit theorem for random walks on symmetric spaces

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年3月24日

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  • Constantin Kogler

摘要

We reduce the local limit theorem for a non-compact semisimple Lie group acting on its symmetric space to establishing that a natural operator associated to the measure is quasicompact. Under strong Diophantine assumptions on the underlying measure, we deduce the necessary spectral results for the operator in question. We thereby give the first examples of finitely supported measures satisfying such a local limit theorem. Moreover, quantitative error rates for the local limit theorem are proved under additional assumptions.