  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.
