On the conservativeness and the recurrence of symmetric jump-diffusions

Sufficient conditions for a symmetric jump-diffusion process to be conservative and recurrent are given in terms of the volume of the state space and the jump kernel of the process. A number of examples are presented to illustrate the optimality of these conditions; in particular, the situation is allowed to be that the state space is topologically disconnected but the particles can jump from a connected component to the other components.Comment: 22 page

H-compactness of elliptic operators on weighted Riemannian Manifolds

In this paper we study the asymptotic behavior of second-order uniformly elliptic operators on weighted Riemannian manifolds. They naturally emerge when studying spectral properties of the Laplace-Beltrami operator on families of manifolds with rapidly oscillating metrics. We appeal to the notion of H-convergence introduced by Murat and Tartar. In our main result we establish an H-compactness result that applies to elliptic operators with measurable, uniformly elliptic coefficients on weighted Riemannian manifolds. We further discuss the special case of ``locally periodic'' coefficients and study the asymptotic spectral behavior of compact submanifolds of $\mathbb R^n$ with rapidly oscillating geometry.Comment: Major revision: In particular, we added various examples and visualization

A note on self-adjoint extensions of the Laplacian on weighted graphs

We study the uniqueness of self-adjoint and Markovian extensions of the Laplacian on weighted graphs. We first show that, for locally finite graphs and a certain family of metrics, completeness of the graph implies uniqueness of these extensions. Moreover, in the case when the graph is not metrically complete and the Cauchy boundary has finite capacity, we characterize the uniqueness of the Markovian extensions.Comment: 17 pages. The assumption of "finite jump size" found in Theorems 1 and 2 in the previous version has been replaced by a weaker condition concerning the newly introduced notion of a "combinatorial neighborhood" in Theorem 1 and has been removed altogether from Theorem 2. Some references added. Final version to appear in J. Funct. Ana