Probabilistic Trace and Poisson Summation Formulae on Locally Compact Abelian Groups

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.