Jun Masamune

Conservation property of symmetric jump processes

Many papers concerning the conservation property for various Markov processes are known so far. In particular, the conservativeness for symmetric diffusion processes is known as giving the volume growth conditions with respect to the ‘intrinsic metric’ relative to the processes. On the other hand, there are few papers for which the conservativeness property is derived in the case of jump processes. In this talk, we will focus on symmetric jump processes and give a condition on the jump (Levy) kernel to derive the conservation property.

Essential Self-Adjointness of a Sublaplacian via Heat Equation

The essential self-adjointness of a sublaplacian is studied. It is shown that if the manifold has pseudo-0 negligible boundary, then the sublaplacian is essentially self-adjoint. Especially if the manifold is complete, then the sublaplacian is essentially self-adjoint.

On an inclusion of the essential spectrum of Laplacians under non-compact change of metric

It is shown the stability of the essential self-adjointness, and an inclusion of the essential spectra of Laplacians under the change of Riemannian metric on a subset K of M . The set K may have infinite volume measured with the new metric and its completion may contain a singular set such as fractal, to which the metric is not extendable.

The Serre duality theorem for a non-compact weighted CR manifold

It is proved that the Hodge decomposition and Serre duality hold on a non-compact weighted CR manifold with negligible boundary. A complete CR manifold has negligible boundary. Some examples of complete CR manifolds are presented.

The Liouville property of unbounded fractal layers

We consider an unbounded fractal layer S of Sierpiński type and we prove some analytic properties such as the essential self-adjointness of the Laplacian, the stochastic completeness of S, and Liouville-type theorems for subharmonic functions.