  The subject of this paper is the study of convolution semigroups of states on
a locally compact quantum group, generalising classical families of
distributions of a L\'{e}vy process on a locally compact group. In particular a
definitive one-to-one correspondence between symmetric convolution semigroups
of states and noncommutative Dirichlet forms satisfying the natural translation
invariance property is established, extending earlier partial results and
providing a powerful tool to analyse such semigroups. This is then applied to
provide new characterisations of the Haagerup Property and Property (T) for
locally compact quantum groups, and some examples are presented. The proofs of
the main theorems require developing certain general results concerning
Haagerup's $L^{p}$-spaces.
