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

A Microlocal Analysis of the L\'evy Generator with Conjugate Points