Levy Operators on Manifolds

学生将在$L^{2}$-框架中研究黎曼流形上作为与迷向Levy过程有关的半群的无穷小生成元出现的线性算子。这些是拉普拉斯-贝尔特拉米算符通过非定域项的微扰，非定域项作为沿测地线跳跃的叠加而产生。特别是，她将寻求(I)算子是自伴的充要条件，并寻求关于它的谱的一些信息。(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)研究半群的升力类似于正交格架丛的性质。并找出这两类半群之间的自然联系。(B)研究$p$形式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.