On growth,recurrence and the Choquet-Deny Theorem for <Emphasis Type="Italic">p</Emphasis>-adic Lie groups |
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Authors: | Email author" target="_blank">C R E?RajaEmail author |
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Institution: | (1) Indian Statistical Institute, Statistics and Mathematics Unit, 8th Mile Mysore Road, Bangalore, 560 059, India |
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Abstract: | We first study the growth properties of p-adic Lie groups and its connection with p-adic Lie groups of type R and prove that a non-type R p-adic Lie group has compact neighbourhoods of identity having exponential growth. This is applied to prove the growth dichotomy
for a large class of p-adic Lie groups which includes p-adic algebraic groups. We next study p-adic Lie groups that admit recurrent random walks and prove the natural growth conjecture connecting growth and the existence
of recurrent random walks, precisely we show that a p-adic Lie group admits a recurrent random walk if and only if it has polynomial growth of degree at most two. We prove this
conjecture for some other classes of groups also. We also prove the Choquet-Deny Theorem for compactly generated p-adic Lie groups of polynomial growth and also show that polynomial growth is necessary and sufficient for the validity of
the Choquet-Deny for all spread-out probabilities on Zariski-connected p-adic algebraic groups. Counter example is also given to show that certain assumptions made in the main results can not be
relaxed. |
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Keywords: | 20F15 22E35 60B15 60J15 60J45 |
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