Properties of random walks on discrete groups: Time regularity and off-diagonal estimates |
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Authors: | Nick Dungey |
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Institution: | Department of Mathematics, Macquarie University, NSW 2109 Australia |
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Abstract: | In this paper we study some properties of the convolution powers K(n)=K∗K∗?∗K of a probability density K on a discrete group G, where K is not assumed to be symmetric. If K is centered, we show that the Markov operator T associated with K is analytic in Lp(G) for 1<p<∞, and prove Davies-Gaffney estimates in L2 for the iterated operators Tn. This enables us to obtain Gaussian upper bounds for the convolution powers K(n). In case the group G is amenable, we discover that the analyticity and Davies-Gaffney estimates hold if and only if K is centered. We also estimate time and space differences, and use these to obtain a new proof of the Gaussian estimates with precise time decay in case G has polynomial volume growth. |
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Keywords: | 60B15 60G50 47D06 |
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