A Microlocal Analysis of the Lévy Generator with Conjugate Points

Abstract

We analyze the microlocal structure of the infinitesimal generator of a Lévy process on a compact, connected Riemannian manifold when conjugate points are allowed. We show that if there are no singular conjugate pairs, then the infinitesimal generator equals two pseudodifferential operators plus an infinite sum of Fourier integral operators. This extends known results for the flat torus, the round sphere, and Anosov manifolds.