Probabilistic Trace and Poisson Summation Formulae on Locally Compact Abelian Groups
Abstract
We investigate convolution semigroups of probability measures with continuous densities on locally compact abelian groups, which have a discrete subgroup such that the factor group is compact. Two interesting examples of the quotient structure are the $d$dimensional torus, and the adèlic circle. Our main result is to show that the Poisson summation formula for the density can be interpreted as a probabilistic trace formula, linking values of the density on the factor group to the trace of the associated semigroup on $L^{2}$space. The Gaussian is a very important example. For rotationally invariant $\alpha$stable densities, the trace formula is valid, but we cannot verify the Poisson summation formula. To prepare to study semistable laws on the adèles, we first investigate these on the $p$adics, where we show they have continuous densities which may be represented as series expansions. We use these laws to construct a convolution semigroup on the adèles whose densities fail to satisfy the probabilistic trace formula.
 Publication:

arXiv eprints
 Pub Date:
 February 2016
 DOI:
 10.48550/arXiv.1602.01252
 arXiv:
 arXiv:1602.01252
 Bibcode:
 2016arXiv160201252A
 Keywords:

 Mathematics  Probability