  We analyze the microlocal structure of the infinitesimal generator of a
L\'evy process on a closed Riemannian manifold when conjugate points are
allowed. We show that if there are no singular conjugate pairs, then the
infinitesimal generator can be written as a sum of pseudodifferential operators
and Fourier integral operators. This extends and unifies known results for the
flat torus, the sphere, and Anosov manifolds.
