A Microlocal Analysis of the Lévy Generator with Conjugate Points

We analyze the microlocal structure of the infinitesimal generator of a Lévy 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 is microlocally equal to a sum of pseudodifferential operators and Fourier integral operators. This provides a microlocal analogue of known results for the flat torus, the sphere, and Anosov manifolds.