Probabilistic Trace and Poisson Summation Formulae on Locally Compact Abelian Groups

TL;DR

This paper explores the connection between Poisson summation, trace formulas, and convolution semigroups of probability measures on locally compact abelian groups, with applications to adèles, p-adics, and stable laws.

Contribution

It introduces a probabilistic interpretation of the Poisson summation formula as a trace formula and investigates its validity for various stable densities on different groups.

Findings

Trace formula holds for Gaussian and rotationally invariant alpha-stable densities.

Constructed convolution semigroup on adèles with densities that do not satisfy the trace formula.

Series expansions for densities on p-adics enable analysis of their properties.

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 ad\`{e}lic 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 ad\`{e}les, 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 ad\`{e}les whose densities fail to satisfy the probabilistic trace formula.