  The behavior of affine processes, which are ubiquitous in a wide range of
applications, depends crucially on the choice of state space. We study the case
where the state space is compact, and prove in particular that (i) no diffusion
is possible; (ii) jumps are possible and enforce a grid-like structure of the
state space; (iii) jump components can feed into drift components, but not vice
versa. Using our main structural theorem, we classify all bivariate affine
processes with compact state space. Unlike the classical case, the
characteristic function of an affine process with compact state space may
vanish, even in very simple cases.
