A microlocal analysis of the Lévy generator with conjugate points (Q6624045)

Lévy processes on Riemannian manifolds are subjects of intensive research, starting from the work \textit{D. Applebaum} and \textit{A. Estrade} [Ann. Probab. 28, No. 1, 166--184 (2000; Zbl 1044.60035)]. In the present paper the author gives a microlocal analysis of the corresponding generator in terms of the pseudo-differential operators and the Fourier integral operators of [\textit{L. Hörmander}, The analysis of linear partial differential operators. IV: Fourier integral operators. Berlin etc.: Springer-Verlag (1985; Zbl 0612.35001)]. Namely, under geometric assumptions on the metric allowing cojugate points with exclusion of singular conjugate pairs, the generator is described as sum of pseudo-differential and Fourier operators. In the case of the Anosov Riemannian metric, conjugate points do not exist and Fourier operators do not appear in the expression. Note also that in two dimension singular conjugate pairs cannot exist, so in this case the result of the author covers all the possibilities.