Levy Operators on Manifolds

学生将在$L^{2}$-框架中研究黎曼流形上与迷向Levy过程相关的半群的无穷小生成元的线性算子。这些是拉普拉斯-贝尔特拉米算子通过非局部项的扰动，这些非局部项作为沿沿着测地线的跳跃的叠加而出现。特别是她将寻求（i）必要和充分条件的运营商是自伴的，并寻求获得一些信息，其频谱。(ii)半群是紧的和迹类的充要条件。(iii)基本过程具有（正则）密度的条件，以及将密度的对角部分与半群的迹联系起来的“迹公式”。她还将通过势项研究生成元的Feynman-Kac型扰动。最后，她应该（a）研究半群到规范正交标架丛的提升的类似性质，并找到两类半群之间的自然关系。(b)研究p型空间上的诱导半群和生成元。

The student will investigate the linear operators which arise as infinitesimal generators of semigroups associated to isotropic Levy processes on Riemannian manifolds in an $L^{2}$-framework. These are perturbations of the Laplace-Beltrami operator through non-local terms that arise as a superposition of jumps along geodesics. In particular she will seek(i) Necessary and sufficient conditions for the operator to be self-adjoint and seek to get some information about its spectrum.(ii) Necessary and sufficient conditions for the semigroup to be compact and trace-class.(iii) Conditions for the underlying process to have a (regular) density, and a "trace formula" which relates the diagonal part of the density to the trace of the semigroup.She will also study Feynman-Kac type perturbations of the generator through a potential term.Finally, she should (a) Study similar properties for lift of the semigroup to the bundle of orthonormal frames, and find natural relationships between the two classes of semigroups. (b) Study the induced semigroups and generators on spaces of $p$ forms.